View source for 744 (number)

{{Infobox number
| number = 744
| divisor = 1, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744
}}

'''744''' ('''seven hundred [and] forty four''') is the [[Natural number|natural number]] following [[743 (number)|743]] and preceding 745.

744 plays a major role within moonshine theory of [[Sporadic group|sporadic group]]s, in context of the [[Classification of finite simple groups|classification of finite simple groups]].

== Number theory ==

'''744''' is the nineteenth number of the form <math>pqr^{3}</math> where <math>r</math>, <math>p</math> and <math>q</math> represent distinct [[Prime number|prime number]]s ([[2|2]], [[3|3]], and [[31 (number)|31]]; respectively).<ref name="pqr3">{{Cite OEIS |A189975 |Numbers with prime factorization pqr^3. |access-date=2023-07-16 }}</ref>

It can be represented as the sum of nonconsecutive [[Factorial|factorial]]s <math>k!</math>,<ref name="N-CF">{{Cite OEIS |A060112 |Sums of nonconsecutive factorial numbers. |access-date=2023-07-16 }}</ref> as the sum of four consecutive primes <math>p</math>,<ref>{{Cite OEIS |A034963 |Sums of four consecutive primes. |access-date=2023-07-16 }}</ref> and as the product of [[Divisor function|sums of divisors]] <math>\sigma(n)</math> of consecutive [[Integer|integer]]s <math>n</math>;<ref>{{Cite OEIS |A083539 |a(n) is sigma(n) * sigma(n+1) as the product of sigma-values for consecutive integers. |access-date=2023-07-16 }}</ref> respectively:{{efn-lr|1= 744 is equal to the sum between the 41st and 44th [[Sequence|indexed]] prime numbers, inclusive. Indices 15 and 16 of '''σ'''(''n'') multiply to 240, which is the Euler totient value of 744,<ref name="Etot">{{Cite OEIS |A000010 |Euler totient function phi(n): count numbers less than or equal to n and prime to n. |access-date=2023-07-16 }}</ref> and add to 31 which is the largest prime that is not a totative of 744 (less than).<br /> A sum between [[24 (number)|24]] and [[31 (number)|31]] generates the tenth [[Triangular number|triangular number]] [[55 (number)|55]], where the eleventh triangular number is '''[[66 (number)|66]]'''.<ref name="TriangNum">{{Cite OEIS |A000217 |Triangular number: a(n) is the binomial(n+1,2) equivalent to n*(n+1)/2 that is 0 + 1 + 2 + ... + n. |access-date=2023-09-02 }}</ref> 121 is the sum between these two triangular numbers, which is equivalent to the [[Square number|square]] of 11. The prime index of 31 and its [[Permutable prime|permutable prime]] in [[Decimal|decimal]] (13)<ref name="PermP">{{Cite OEIS |A003459 |Absolute primes (or permutable primes): every permutation of the digits is a prime. |access-date=2023-12-14 }}</ref> form the third pair of [[Twin prime|twin prime]]s {{math|1=(11, 13)}},<ref name="TwinPs">{{Cite OEIS |A001359 |Lesser of twin primes. |access-date=2023-10-18 }}</ref> whose sum is 24, with respective prime indices 5 and 6<ref name="PNum">{{Cite OEIS |A000040 |The prime numbers. |access-date=2023-09-06 }}</ref> that add to 11. }} 

<math display=block>
\begin{align}
744 & = 4! + 6! \\
744 & = 179 + 181 + 191 + 193 \\
744 & = \sigma(15) \times \sigma(16) = 24 \times 31 \\
\end{align}</math>

744 contains sixteen total [[Divisor|divisor]]s — fourteen aside from its largest and smallest [[Unitary divisor|unitary divisor]]s — all of which [[Arithmetic number|collectively generate]] an integer [[Arithmetic mean|arithmetic mean]] of <math>120 = 5!</math><ref name="ArithNum0">{{Cite OEIS |A003601 |Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n). |access-date=2023-07-16 }}</ref><ref name="ArithNum">{{Cite OEIS |A102187 |Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer). |access-date=2023-07-16 }}</ref> that is also the first number of the form <math>pqr^{3}.</math><ref name="pqr3" />{{efn-lr|1='''[[120 (number)|120]]''' is also equal to the sum of the first fifteen integers, or fifteenth [[Triangular number|triangular number]] {{math|1='''Σ'''{{su|b=<sub>''n''{{hair space}}={{hair space}}1</sub>|p=<sub>15</sub>}}{{hair space}}''n'' }},<ref name="TriangNum" /> while it is also the smallest number with sixteen divisors,<ref name="LeastD">{{Cite OEIS |A005179 |Smallest number with exactly n divisors. |access-date=2023-11-06 }}</ref> with 744 the thirty-first such number.<ref>{{Cite OEIS |A000005 |d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n. |access-date=2023-12-28 }}</ref> This value is also equal to the sum of all the prime numbers less than 31 that are not [[Integer factorization|factors]] of 744 except for 5, and including 1. Inclusive of 5, this sum is equal to {{math|1=[[125 (number)|125]] = 5<sup>3</sup> }}, which is the second number after [[32 (number)|32]] to have an [[Aliquot sum|aliquot sum]] of 31.<ref name="AliSum">{{Cite OEIS |A001065 |Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n. |access-date=2023-09-02  }}</ref><br />In the [[Collatz conjecture]], 744 and 120 both require fifteen steps to reach [[5|5]], before cycling through {16, 8, 4, 2, 1} in five steps.<ref name="3x+1">{{Cite web |editor-last=Sloane |editor-first= N. J. A. |url=http://oeis.org/wiki/3x%2B1_problem |title=3x+1 problem |website=The [[On-Line Encyclopedia of Integer Sequences]] |publisher=The OEIS Foundation |access-date=2023-09-22 }}</ref><ref name="ColOrb">{{Cite OEIS |A006577 |Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached. |access-date=2023-09-22 }}</ref> Otherwise, they both require '''nineteen''' steps to reach '''2''', which is the middle node in the {1,4,2,1,4...} elementary trajectory for 1 when cycling back to itself, or '''twenty''' steps to reach '''1'''. }}

The number ''[[Partition (number theory)|partitions]]'' of the [[Square number|square]] of [[7|seven]] ([[49 (number)|49]]) into prime parts is 744,<ref name="PartNPP">{{Cite OEIS |A000607 |Number of partitions of n into prime parts. |access-date=2023-07-16 }}</ref> as is the number of partitions of [[48 (number)|48]] into at most four distinct parts.<ref>{{Cite OEIS |A347586 |Number of partitions of n into at most 4 distinct parts. |access-date=2023-07-16 }}</ref>{{efn-lr|1=The '''radical''' {{math|1=186 = 2 × 3 × 31 }} of 744<ref name="RadN">{{Cite OEIS |A007947 |Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n. |access-date=2023-12-31 }}</ref> has an [[Arithmetic mean|arithmetic mean]] of divisors equal to 48,<ref name="ArithNum" /> where the sum between the three distinct prime factors of 744 is {{math|1=6<sup>2</sup> = [[36 (number)|36]] }}. [[186 (number)|186]] is [[Nontotient|nontotient]]<ref name="nontot" /> and [[Noncototient|noncototient]],<ref>{{Cite OEIS |A005278 |Noncototients: numbers k such that x - phi(x) equal to k has no solution. |access-date=2023-09-24 }}</ref> equal to the number of pentahexes when [[Rotational symmetry|rotations]] are counted as distinct.<ref>{{cite OEIS |A001207 |Number of fixed hexagonal polyominoes with n cells. |access-date=2023-09-24 }}</ref> }}

744 is an [[Abundant number|abundant number]],<ref>{{Cite OEIS |A005101 |Abundant numbers (sum of divisors of m exceeds 2m). |access-date=2023-04-03 }}</ref> with an ''abundance'' of 432.<ref name="AbunoN">{{Cite OEIS |A033880 |Abundance of n, or (sum of divisors of n) - 2n. |access-date=2023-12-29 }}</ref>{{efn-lr|1=744 is the 181st abundant number. Specifically, 432 (twice [[216 (number)|216]], the cube of [[6|6]]) has an aliquot sum of [[800 (number)|808]],<ref name="AliSum" /> where 433 is the thirty-first ''full repetend prime'' in [[Decimal|decimal]] that repeats 432 digits, which collectively add to 1944.<ref name="FullRepP">{{Cite OEIS |A001913 |Full reptend primes: primes with primitive root 10. |access-date=2023-12-15 }}</ref> Less than 1000, the sixtieth and largest such prime number is 983, which repeats 982 digits that sum to 4419, a two-digit dual permutation of the digits of 1944 (and where {{math|1=2475 = 4419 − 1944 }} has a reduced totient of 60).<ref name="redTot" /> 432 is also the twenty-third [[Interprime|interprime]] between [[Twin prime|twin prime]]s,<ref name="InterP">{{Cite OEIS |A014574 |Average of twin prime pairs. |access-date=2023-12-30 }}</ref> and the eighth such number whose adjacent prime numbers have prime indices that add to a prime number ({{math|1=[[167 (number)|167]] = 83 + 84 }}, the thirty-ninth prime);<ref name="PNum" /> 348, the composite index of 432,<ref name="CompNum" /> is itself the sixth interprime whose adjacent prime numbers have indices that add to a prime number ({{math|1=139 = 69 + 70 }}, the thirty-fourth prime).<ref name="PNum" /><br />Importantly, {{math|1=432 = 3<sup>3</sup> + 4<sup>3</sup> + 5<sup>3</sup> + 6<sup>3</sup> }}<ref>{{Cite OEIS |A217843 |Numbers which are the sum of one or more consecutive nonnegative cubes. |access-date=2023-12-30 }}</ref> is an [[Achilles number]], a [[Powerful number|powerful number]] (the thirty-fourth)<ref>{{Cite OEIS |A001694 |Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers). |access-date=2023-12-30 }}</ref> that is itself not a [[Perfect power|perfect power]], like 864 (the eleventh index, that is twice its value), and [[288 (number)|288]] (the totient of 864),<ref name="Etot" /> as well as [[108 (number)|108]] (a fourth of 432), ''and'' '''1944'''.<ref>{{Cite OEIS |A052486 |Achilles numbers - powerful but imperfect: if n equal to Product(p_i^e_i) then all e_i less than 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power. |access-date=2023-12-29 }}</ref> (Note, that 432 is the sixth member, where the sixth [[Triangular number|triangular number]] is [[21 (number)|21]],<ref name="TriangNum" /> itself the index of 1944 in the list of Achilles numbers. Notice also that {{math|1=1944 = '''1200''' + 744 }}, where {{math|1=1200 = 456 + 744 }}, or in other words {{math|1='''σ'''<sub></sub>(456)}}.)<ref name="SigmaSoD" /><br /> On the other hand, '''[[181 (number)|181]]''', the index of 744 as an abundant number, is the sixteenth full repetend prime in base ten,<ref name="FullRepP" /> and the composite index of [[232 (number)|232]],<ref name="CompNum" /> that is the sum of the first five interprimes to lie in-between twin primes with prime sums from respective indices (i.e., 138, 72, 12, 6, 4), as with 432 and 348.<ref name="InterP" /> [[109 (number)|109]] is the tenth full repetend prime, repeating '''108''' digits whose sum of repeating digits is 486, a fourth of 1944, where 487 is the thirty-third full repetend prime (and 811 the forty-ninth, one less than the sum of the '''180''' repeating digits of 181; with {{math|1=288 = 180 + 108 }}).<ref name="FullRepP" /> }} It is [[Semiperfect number|semiperfect]], since it is equal to the sum of a subset of its divisors (e.g., 1 + 2 + 4 + 24 + '''62''' + 93 + 124 + 186 + 248).<ref>{{Cite OEIS |A005835 |Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n. |access-date=2023-04-03  }}</ref>{{efn-lr|1=744 is the 183rd semiperfect number, an index value whose sum-of-divisors is '''248''',<Ref name="SigmaSoD" /> and an arithmetic mean of divisors equal to {{math|1='''[[62 (number)|62]]''' = 31 × 2 }},<ref name="ArithNum0" /><ref name="ArithNum" /> both of which are divisors of 744 (fourteenth  and tenth largest, respectively); otherwise, the sum of its two divisors greater than 1 is {{math|1=3 + 61 = 64 = 8<sup>2</sup> }}. '''183''' is also the ''eighth'' [[Perfect totient number|perfect totient number]],<ref name="PerfTot" /> and the number of [[Semiorder|semiorder]]s on four labeled elements.<ref>{{Cite OEIS |A006531 |Semiorders on n elements. |access-date=2024-01-02 }}</ref> It is the largest number of interior regions formed by fourteen intersecting circles, which is equivalent to the number of points in the [[Projective plane|projective plane]] over the [[Finite field|finite field]] {{math|1=<math>\mathbb{Z}</math><sub>13</sub> }}, since {{math|1=13<sup>2</sup> + 13 + 1 = 183}}.<ref>{{Cite OEIS |A002061 |Central polygonal numbers: a(n) equal to n^2 - n + 1. |access-date=2024-01-02 }}</ref> It is the number of toothpicks in the ''toothpick sequence'' after eighteen stages.<ref>{{Cite OEIS |A139250 |Toothpick sequence (see Comments lines for definition). |access-date=2024-01-02 }}</ref> Following, 181, 183 is the sixty-second [[Löschian number]] of the form {{math|1=''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup> }}, as a product of the third and twenty-fourth such numbers ([[3|3]], [[61 (number)|61]]), of prime indices two and eighteen.<ref>{{Cite OEIS |A003136 |Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice. |access-date=2023-01-02 }}</ref><ref name="PNum" /> Furthermore, the smallest number to have exactly four solutions to this [[Quadratic function|quadratic polynomial]] is the first '''[[Taxicab number|taxicab number]]''' '''[[1729 (number)|1729]]''', from {{math|1=(''a'',''b'') }} integer pairs {{math|1=(3, 40), (8, 37), (15, 32) }} and {{math|1=(23, 25) }},<ref>{{Cite web |last=Mitchell |first=David |title=Tessellating the Ramanujan-Hardy Taxicab Number, 1729, Bedrock of Integer Sequence A198775 |website= 
LatticeLabyrinths |url=https://latticelabyrinths.wordpress.com/2017/02/25/tessellating-the-hardy-ramanujan-taxicab-number-1729-bedrock-of-integer-sequence-a198775/  |access-date=19 July 2018 |date=February 25, 2017 }}</ref><ref>{{Cite OEIS |A198775 |Numbers having exactly four representations by the quadratic form x^2+xy+y^2 with 0 less than or equal to x less than or equal to y. |access-date=2024-01-03 }}</ref> where these four pairs collectively generate a sum of '''183'''. The smallest such number with only two solutions, on the other hand, is '''49''';<ref>{{Cite OEIS |A198773 |Numbers having exactly two representations by the quadratic form x^2+xy+y^2 with 0 less than or equal to x less than or equal to y. |access-date=2024-01-03 }}</ref> wherein 1729 is the 97th such number expressible in two, or more, ways.<ref>{{Cite OEIS |A118886 |Numbers expressible as x^2 + x*y + y^2, 0 less than or equal to x less than or equal to y in 2 or more ways. |access-date=2024-01-03 }},</ref> The probability that any odd number is a Löschian number is '''0.75''', while the probability that it is an even number is a remaining 0.25.<ref>{{Cite journal |last=Marshall |first=John U. |title=The Loeschian numbers as a problem in number theory |journal=Geographical Analysis |url=https://oeis.org/A003136/a003136_2.pdf |volume=7 |issue=4 |year=1975 |pages=421–426 |doi=10.1111/j.1538-4632.1975.tb01054.x |doi-access=free |bibcode=1975GeoAn...7..421M |via=[[On-Line Encyclopedia of Integer Sequences|OEIS]] }}</ref><br /> There are precisely '''816''' integers (uninclusively, equal to the sum-of-divisors of '''737''', the largest composite totative of 744)<ref name="SigmaSoD" /> between the semiperfect index of 744 and 1000, a number that in-turn has a sum-of-divisors of {{math|1=[[2000 (number)#2200 to 2299|2232]] = 744 × 3 }},<ref name="SigmaSoD" /> the 24th decagonal number,<ref>{{Cite OEIS |A001107 |10-gonal (or decagonal) numbers: a(n) is n*(4*n-3). |access-date=2023-01-03 }}</ref> as well as the 30th number that can be expressed as the difference of the squares of consecutive primes in just ''one'' distinct way;<ref name="DoSCP">{{Cite OEIS |A090785 |Numbers that can be expressed as the difference of the squares of consecutive primes in just one way. |access-date=2023-01-03 }}</ref> other members in this sequence include: {{Bullet list |'''1920''' ('''26'''th), the sum-of-divisors of 744,<ref name="SigmaSoD" /> |[[1728 (number)|1728]] (24th), equal to the cube of 12, and one less than the first taxicab number, |'''432''' ('''8'''th), the abundance of 744,<ref name="AbunoN" /> |288 (6th), the difference between 744 and '''456''', the only number to have an aliquot sum of 744,<ref name="AliSum" /> |'''240''' ('''5'''th), the Euler totient of 744.<ref name="Etot" /> }} '''[[888 (number)|888]]''' is the '''13'''th member, the 26th repdigit in decimal<ref name="Repd" /> equal to the sum between 432 and 456, and between 144 and 744 (where 432 and 456 both have totient values of 144);<ref name="Etot" /> and equal to the product between the twelfth prime number 37, and 24. The first four members (5, 16, 24, 48) generate a sum of [[93 (number)|93]], which is the ''eleventh'' largest divisor of 744, following 62.<ref name="DoSCP" /> 93 represents the number of different [[Cyclic permutation|cyclic]] Gilbreath permutations on '''11''' elements,<ref>{{Cite OEIS|A000048 |Number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged. |access-date=2023-01-03 }}</ref> and consequently there are ninety-three different [[Real number|real]] [[Periodic point|periodic point]]s of order 11 on the [[Mandelbrot set]].<ref>{{Cite book |first1=Persi |last1=Diaconis |first2=Ron |last2=Graham |title=Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks |chapter=Chapter 5: From the Gilbreath Principle to the Mandelbrot Set |chapter-url=http://assets.press.princeton.edu/chapters/s5_9510.pdf |publisher=Princeton University Press |year=2011 |pages=61–83 |doi=10.1515/9781400839384-007 |isbn=978-1-4008-3938-4 |mr=2858033 }}.</ref>
Otherwise, {{math|1='''1632''' = 1176 + 456 }}, in equivalence with the aliquot sum of 744 (1176)<Ref name="AliSum" /> and 456, is the sixteenth number that can be expressed as the difference of the [[Square number|squares]] of primes in just ''two'' distinct ways, consecutive or otherwise (432, 1728, 1920 and 2232 are also in this sequence).<ref name="DoSP">{{Cite OEIS |A090788 |Numbers that can be expressed as the difference of the squares of primes in just two distinct ways. |access-date=2024-01-03 }}</ref><hr> 183 is also the first non-trivial '''62'''-gonal number. }}

It is also a [[Practical number|practical number]],<ref name="PractNum">{{Cite OEIS |A005153 |Practical numbers |access-date=2023-04-03 }}</ref> and the first number to be the sum of nine [[Cube (algebra)|cubes]] in eight or more ways,<ref>{{Cite OEIS |A345547 |Numbers that are the sum of nine cubes in eight or more ways. |access-date=2023-07-16 }}</ref> as well as the number of six-digit [[Perfect power|perfect power]]s in [[Decimal|decimal]].<ref>{{Cite OEIS |A075308 |Number of n-digit perfect powers. |access-date=2023-07-16 }}</ref>

Meanwhile, in [[Radix|septenary]] 744 is palindromic (2112<sub>7</sub>),<ref name="PalindB7">{{Cite OEIS |A029954 |Palindromic in base 7. |access-date=2023-12-30 }}</ref> while in [[Binary number|binary]] it is a [[Pernicious number|pernicious number]], as its [[Numerical digit|digit representation]] (1011101000<sub>2</sub>) contains a prime count (5) of ones.<ref>{{Cite OEIS |A052294 |Pernicious numbers: numbers with a prime number of 1's in their binary expansion. |access-date=2023-08-28 }}</ref>{{efn-lr|1=744 is the four hundred and sixth indexed pernicious number, where '''406''' is the twenty-eighth triangular number;<ref name="TriangNum" /> in its base-two representation, the [[Positional notation|digit positions]] of zeroes are in {{math|1=1:1:3 }} or {{math|1=3:1:1 }} ratio with the positions of ones, which are in {{math|1=1:3:1 }} ratio.<br />Its [[Ones' complement|ones' complement]] is 100010111<sub>2</sub>, equivalent to {{math|1=[[279 (number)|279]] = 3<sup>2</sup> × 31 }} in [[Decimal|decimal]], which represents the sum of GCDs of parts in all partitions of {{math|1='''[[16 (number)|16]]''' = 4<sup>2</sup> }}.<ref>{{Cite OEIS |A078392 |Sum of GCD's of parts in all partitions of n. |access-date=2023-10-01 }}</ref> It is also the number of partitions of {{math|1=[[62 (number)|62]] = 2 × 31 }} (a divisor of 744) as well as [[63 (number)|63]] into factorial parts (without including 0!),<ref>{{Cite OEIS |A064986 |Number of partitions of n into factorial parts (0! not allowed). |access-date=2023-10-01 }}</ref> and the number of integer partitions of [[44 (number)|44]] whose length is equal to the [[Least common multiple|LCM]] of all parts<ref>{{Cite OEIS |A316433 |Number of integer partitions of n whose length is equal to the LCM of all parts. |access-date=2023-10-01 }}</ref> (with 63 the forty-fourth [[Composite number|composite number]],<ref name="CompNum">{{Cite OEIS |A002808 |The composite numbers: numbers n of the form x*y for x > 1 and y > 1. |access-date=2023-09-27 }}</ref> where 44 is itself the number of [[Derangement|derangement]]s of 5,<ref>{{Cite OEIS |A000166 |Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points. |access-date=2023-10-01 }}</ref> and {{math|1=63 + 44 = [[107 (number)|107]] }} the twenty-eighth [[Prime number|prime number]]).<ref name="PNum" /><hr>'''2112'''<sub>10</sub> is the number of repeating decimal digits of 2113 as a full repetend prime in ''decimal'',<ref name="FullRepP" /> where 2112 is the 65th ''interprime'' to lie between consecutive twin primes,<ref name="InterP" /> otherwise it is the '''317'''th member between consecutive odd primes,<ref name="AvgCP">{{Cite OEIS |A024675 |Average of two consecutive odd primes. |access-date=2023-12-30 }}</ref> where 317 is the 66th indexed prime number (with 90 and 91 the 65th and 66th composites, respectively, where {{math|1=181 = 90 + 91 }}).<ref name="PNum" /><ref name="CompNum" /> '''279''', on the other hand, is the 58th number to lie between consecutive odd prime numbers ([[277 (number)|277]], [[281 (number)|281]]),<ref name="AvgCP" /> with {{math|1='''81''' = 9<sup>2</sup> }} the fifty-eighth composite number.<ref name="CompNum" />{{efn-lr|1=Meanwhile, where 279 is the 58th interprime to lie between consecutive odd numbers that are not necessarily twin primes (here at a difference of four), the 58th palindromic number in septenary is the equivalent of 456 in decimal<ref name="PalindB7" /> (that is the only number to have an aliquot sum of 744)<ref name="AliSum" /> and represented as '''1221'''<sub>7</sub>, whose digits are the dual permutation of the digits of 744 as the '''64'''th palindrome in base-7 ('''2112''').<ref name="PalindB7" /><br />Explicitly, every positive integer is at most the sum of ''two hundred and seventy-nine'' [[Eighth power|eighth power]]s ([[Waring's problem]]), preceded by a maximum number of {{math|1={1, 4, 9, 19, 37, 73, 143} }} {{math|1=''n'' }}-powers;<ref>{{Cite OEIS |A002804 |(Presumed) solution to Waring's problem: g(n) equal to 2^n + floor((3/2)^n) - 2. |access-date=2023-10-01 }}</ref> where, as aforementioned, 279 is the decimal representation of the ''ones' complement'' of 744 in binary. In this sequence, the first six members generate a sum equivalent to the seventh member [[143 (number)|143]], the composite index of 186<ref name="CompNum" /> that is the '''radical''' of 744,<ref name="RadN" /> as well as equal to the sum of seven consecutive primes starting from 11 through the eleventh prime number: {{math|1=11 + 13 + 17 + 19 + 23 + 29 + 31 }}. It is also the product between the third [[Twin prime|twin prime]] pair {{math|1=(11 × 13) }};<ref name="TwinPs" /> while figuring in {{math|1=3<sup>4</sup> + 4<sup>4</sup> + 5<sup>4</sup> + 6<sup>4</sup> = 7<sup>4</sup> − 143 }}, which is the second (or third) exception in the sequence of polynomials that starts with {{math|1=3<sup>2</sup> + 4<sup>2</sup> = 5<sup>2</sup> }} and {{math|1=3<sup>3</sup> + 4<sup>3</sup> + 5<sup>3</sup> = 6<sup>3</sup> }}, after {{math|1=3<sup>1</sup> = 4<sup>1</sup> − 1 }} (and, {{math|1=0 = 3<sup>0</sup> − 1 }}).<br />While in the beginning of the sequence of full repetend primes (with [[Primitive root modulo n|primitive root]] 10) the two smallest such numbers are [[7|7]] and [[17 (number)|17]], with the repeating digits of the reciprocal of 7 adding to {{math|1=27 = 3<sup>3</sup> }} that is the seventeenth composite number,<ref name="CompNum" /> there is also {{math|1=983 + 17 = 1000 }}, and {{math|1=2 × 279 + 432 = 7 + 983 }}. The span of full repetend primes between 17 and 983 is of fifty-nine integers, where [[59 (number)|59]] is the seventeenth prime number, and seventh ''[[Super-prime|super-prime]]''.<ref name="PNum" /><ref name="SupP" /> }} More specifically, eighty-one is the sum of the repeating digits of the third full repetend prime in decimal, [[19 (number)|19]],<ref name="FullRepP" /> where the sum of these digits is the '''''magic constant''''' of an {{math|1=18 × 18}} non-normal yet full [[Prime reciprocal magic square|prime reciprocal magic square]] based on its [[Multiplicative inverse|reciprocal]] {{large|(}}{{math|1='''{{sfrac|1|19}}''' }}{{large|)}}.<ref name="WSymes">{{Cite book|last=Andrews |first=William Symes |title=Magic Squares and Cubes |url=http://djm.cc/library/Magic_Squares_Cubes_Andrews_edited.pdf |publisher=[[Company:Open Court Publishing Company|Open Court Publishing Company]] |location=Chicago, IL |year=1917 |pages=176, 177 |isbn=9780486206585 |oclc=1136401 |zbl=1003.05500 |mr=0114763 }}</ref><ref>{{Cite OEIS |A021023 |Decimal expansion of 1/19. |access-date=2023-12-30 }}</ref><ref name="FPRMS10">{{Cite OEIS |A072359 |Primes p such that the p-1 digits of the decimal expansion of k/p (for k equal to 1,2,3,...,p-1) fit into the k-th row of a magic square grid of order p-1 |access-date=2023-12-30 }}</ref> }} 

=== Totients ===

744 has two hundred and forty integers that are relatively prime or ''coprime'' with and up to itself, equivalently its Euler totient.<ref name="Etot" />{{efn-lr|1=744 is the twenty-third of thirty-one such numbers to have a '''totient''' of 240, after 738, and preceding 770. The smallest is [[241 (number)|241]], the fifty-third prime number and sixteenth [[Super-prime|super-prime]],<ref name="SupP">{{Cite OEIS |A006450 |Prime-indexed primes: primes with prime subscripts. |access-date=2023-10-18 }}</ref> and the largest is [[1000 (number)#1001 to 1099|1050]], which represents the number of parts in ''all partitions'' of [[29 (number)|29]] into distinct parts.<ref>{{Cite OEIS |A015723 |Number of parts in all partitions of n into distinct parts. |access-date=2023-12-20 }}</ref> }}

This totient of 744 is [[Regular number|regular]] like its [[Divisor function#Definition|sum-of-divisors]], where 744 sets the twenty-ninth record for <math>\sigma(n),</math> of 1920.<ref name="RecSigm">{{Cite OEIS |A034885 |Record values of sigma(n). |access-date=2023-07-16 }}</ref>{{efn-lr|1=The value of this sigma function represents the fifteenth sum of ''non-triangular'' numbers in-between [[Triangular number|triangular number]]s; in this instance it is the sum that lies in-between the fifteenth ([[120 (number)|120]]) and sixteenth ([[136 (number)|136]]) triangular numbers<ref>{{Cite OEIS |A006002 |a(n) equal to n*(n+1)^2/2 (Sum of the nontriangular numbers between successive triangular numbers. 1, (2), 3, (4, 5), 6, (7, 8, 9), 10, (11, 12, 13, 14), 15, ...) |access-date=2023-08-27 }}</ref> (i.e. the sum of {{math|1=121 + 122 + ... + 135 }}). }} Both the totient and sum-of-divisors values of 744 contain the same set of distinct prime factors ([[2|2]], [[3|3]], and [[5|5]]),<ref>{{Cite OEIS |A081377 |Numbers n such that the set of prime divisors of phi(n) is equal to the set of prime divisors of sigma(n). |access-date=2023-07-16 }}</ref> while the [[Carmichael function]] or ''reduced totient'' (which counts the [[Least common multiple|least common multiple]] of [[Group theory|order]] of elements in a [[Multiplicative group|multiplicative group]] of integers [[Modulo|modulo]] <math>n</math>) at seven hundred forty-four is equal to <math>\lambda(744) = 30 = 2 \times 3 \times 5</math>.<ref name="redTot" />{{efn-lr|1=The thirtieth triangular number is {{math|1=465 = 3 × 5 × 31 }}, equal to the difference between 744 and 279 (equivalently 1011101000<sub>2</sub> and 100010111<sub>2</sub>, ones' complement pairs respectively). 30 is the twelfth number {{math|1=''m'' }} such that {{math |1=6''m'' + 1 }} and {{math|1=6''m'' − 1 }} are twin primes ([[181 (number)|181]], [[179 (number)|179]]).<ref name="TwinPs" /><ref>{{Cite OEIS |A002822 |Numbers m such that 6m-1, 6m+1 are twin primes. |access-date=2023-10-02 }}</ref> 465 is also the binomial {{math|1=(31,2) }} equal to the number of size-2 subsets of {{math|1={0, 1, ..., 31} }} that contain no consecutive integers (in light as a triangular number, trailing its sequence by one index).<ref>{{Cite OEIS |A161680 |a(n) is the binomial(n,2): number of size-2 subsets of {0,1,...,n} that contain no consecutive integers. |access-date=2023-10-02 }}</ref><ref name="TriangNum" /> In the [[Padovan sequence]], 465 is the twenty-eighth indexed member equal to the number of ''[[Composition (combinatorics)|compositions]]'' of [[28 (number)|28]] into parts congruent to {{math|1=2 mod 3 }}, also the number of compositions of 28 into parts that are odd and greater than or equal to 3, and the number of maximal cliques in a {{math|1=(28 + 6) }}–path [[Complement graph|complement graph]] (specifically, in the {{math|1=34}}–path).<ref>{{Cite OEIS |A000931 |Padovan sequence (or Padovan numbers)... |access-date=2023-10-02 }}</ref><hr>'''30''' is the smallest [[Sphenic number|sphenic number]] that is the product of three distinct prime numbers (2, 3, and 5), with the next two such numbers {{math|1=[[42 (number)|42]] + [[66 (number)|66]] = '''108''' }}.<ref>{{Cite OEIS |A007304 |Sphenic numbers: products of 3 distinct primes. |access-date=2024-01-08 }}</ref> Importantly, 30 is the '''''unique point''''' in the sequence of [[Natural number|natural number]]s {{math|1=<math>\mathbb{N}</math><sup>+</sup> }} where the ratio of prime numbers to non-primes (up to) is '''{{sfrac|1|2}}'''; when including [[0|0]] in {{math|1=<math>\mathbb{N}</math><sup>+</sup> '''∪''' {0} = <math>\mathbb{N}</math><sub>0</sub> }} as a non-prime, this ratio is reached at '''29''', where a ratio of one-half is only reached again (here, purely as a ratio between primes and composites) at [[37 (number)|37]] and [[43 (number)|43]], which are respectively the twelfth and fourteenth primes (when including 1 as a ''prime unit'', then the thirteenth prime number [[41 (number)|41]] would be another point of 1:1 ratio, without taking 0 into account).<ref name="PNum" /> Otherwise, a 1:1 ratio of ''primes to composites'' (up to) occurs at two points: 11 and [[13 (number)|13]], respectively the fifth and sixth prime<ref name="PNum" /> — whose indices sum to 11, the prime index of 31 — ''and'' at [[7|7]], when including 0 and 1 as non-primes in {{math|1=<math>\mathbb{N}</math><sub>0</sub> }}, or at [[8|8]] when only including 1, and at [[9|9]] when considering 0 as non-prime and 1 as a prime unit). Thirty is also the first composite value with three distinct prime factors or more for which the Möbius function returns '''[[−1|−1]]''', as with all numbers with a strictly odd number of prime factors, including prime numbers themselves. Relatedly, '''31''' is the first number (after [[1|1]]) to reach a record amplitude between zeros in the associated [[Mertens function]] {{math|1=''M''(''n'') }}, where the next such number is [[114 (number)|114]],<ref>{{Cite OEIS |A304241 |Indices where Mertens function A002321 reaches record amplitudes between zeros. |access-date=2024-01-08 }}</ref> whose arithmetic mean of divisors is 30,<ref name="ArithNum0" /><ref name="ArithNum" /> with a sum-of-divisors of 240.<ref name="SigmaSoD" /> On the other hand, 456 (the only number to have an aliquot sum of 744),<ref name="AliSum" /> 744 and 1176 (the aliquot sum of 744) all set a value of '''−5''' for {{math|1=''M''(''n'') }},<ref>{{Cite OEIS |A002321 |Mertens's function: Sum_{k equal to 1..n} mu(k), where mu is the Moebius function. |access-date=2024-01-08 }}</ref> respectively the 20th, 57th, and 86th numbers to do so (where 57 is the fortieth composite, and 80 the fifty-seventh),<ref name="CompNum" /> with these indices generating the sum {{math|1=20 + 57 + 86 = '''163''' }}, in equivalence with the largest of nine [[Heegner number]]s. (Also, {{math|1=20 + 57 = '''[[77 (number)|77]]''' }}, the sum of [[Decimal|base ten]] digits in the [[Almost integer|almost integer]] approximation of {{math|1=e<sup>{{pi}}√163</sup> }} before its decimal expansion.)<hr>The sixth pair of numbers to generate a record '''''prime gap''''' (of '''14''') are the 30th and 31st prime numbers ([[113 (number)|113]], [[127 (number)|127]]),<ref name="RecPG">{{Cite OEIS |A005250 |Record gaps between primes. |access-date=2024-01-11 }}</ref><ref name="IoPG">{{Cite OEIS |A005669|Indices of primes where largest gap occurs. |access-date=2024-01-11 }}</ref><ref name="PG">{{Cite OEIS |A001223 |Prime gaps: differences between consecutive primes. |access-date=2024-01-11 }}</ref><ref name="PNum" /> The previous consecutive primes to generate record gaps are (89, 97), (23, 29), (7, 11), (5, 3) and (3, 2);<ref name="IoLPGP">{{Cite OEIS |A002386 |Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k. |access-date=2024-01-11 }}</ref><ref name="IoGPGP">{{Cite OEIS |A000101 |Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps). |access-date=2024-01-11 }}</ref> the lesser of these together generate a sum of 127, the thirty-first prime number. }} 744 is also a ''Zumkeller number'' whose divisors can be partitioned into two disjoint [[Set (mathematics)|sets]] with equal sum (960).<ref name="Zumk">{{Cite OEIS |A083207 |Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum. |access-date=2023-07-16 }}</ref>{{efn-lr|1=It is the '''168'''th indexed Zumkeller number, where {{math|1=[[168 (number)|168]] = 6 × 28 }} represents the product of the first two [[Perfect number|perfect number]]s,<ref name="PerfN" /> equal to the fifth '''[[Dedekind number]]''',<ref>{{Cite OEIS |A000372 |Dedekind numbers or Dedekind's problem: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families. |access-date=2023-12-30 }}</ref> where the previous four indexed members (2, 3, 6, 20) collectively add to 31 — with [[496 (number)|496]] the thirty-first triangular number,<ref name="TriangNum" /> and ''third'' perfect number. 168 is also the number of primes below [[1000 (number)|1000]].<ref name="PNum" /><ref>{{Cite OEIS |A006880 |Number of primes less than 10^n. |access-date=2023-12-30 }}</ref> The two sets of divisors of 744 with equal sums are:{{Bullet list |(1, 2, 3, 4, 8, 12, 62, 124, 744) |(6, 24, 31, 186, 93, 248, 372) }} 960 is also the thirty-first ''[[Jordan–Pólya number]]''  that is the product of factorials {{math|1=5! × (2!)<sup>3</sup> }},<ref>{{Cite OEIS |A001013 |Jordan-Polya numbers: products of factorial numbers A000142. |access-date=2023-11-12 }}</ref> equal to the sum of six consecutive prime numbers {{math|1=149 + 151 + 157 + 163 + 167 + 173}}, between the 35th and 40th primes (it is the thirty-fifth such number).<ref>{{Cite OEIS |A127333 |Numbers that are the sum of 6 consecutive primes. |access-date=2023-08-27 }}</ref> The fifteenth and sixteenth triangular numbers generate the sum {{math|1=120 + 136 = [[256 (number)|256]] = 2<sup>8</sup>}} that is the totient value of 960,<ref name="Etot" /> and the number of partitions of {{math|1=[[29 (number)|29]] = 1 + 2 + 3 + 5 + 7 + 11 = 2<sup>2</sup> + 3<sup>2</sup> + 4<sup>2</sup> }} into distinct parts and odd parts.<ref>{{Cite OEIS |A000009 |Expansion of Product_{m greater than or equal to 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts. |access-date=2023-12-15 }}</ref> Like its twin prime 31, 29 is a [[Primorial prime|primorial prime]], which together comprise the third and largest of three known pairs of twin primes (1st, 2nd, and 5th) to be primorial primes,<ref>{{Cite OEIS |A228486 |Near primorial primes: primes p such that p+1 or p-1 is a primorial number (A002110). |access-date=2023-12-16 }}</ref> with possibly no larger pairs. In between these lies '''30'''. }}

Of these [[240 (number)|240]] totatives, 110 are strictly composite totatives that nearly match the sequence of [[Composite number|composite number]]s up to 744 that are [[Modular arithmetic#Congruence|congruent]] to <math>\pm {1} \bmod {6}</math>, which is the same congruence that all prime numbers greater than [[3|3]] hold.<ref>{{Cite OEIS |A038509 |Composite numbers congruent to +-1 mod 6. }}</ref>{{efn-lr|1=The composite totatives are<br />{'''25''', 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 161, 169, 175, 185, 187, 203, 205, 209, 215, 221, 235, 245, 247, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 343, 355, 361, 365, 371, 377, 385, 391, 395, 407, '''413''', '''415''', 425, 427, 437, 445, 451, 455, 469, 473, 475, 481, 485, 493, 497, 505, 511, 515, 517, 529, 533, 535, 539, 545, 551, 553, 559, 565, 575, 581, 583, 595, 605, 611, 623, 625, 629, 635, 637, 649, 655, 665, 667, 671, 679, 685, 689, 695, 697, 703, 707, 715, 721, 725, 731, '''737'''}.<hr>Its smallest composite totative is {{math|1=5<sup>2</sup> = '''[[25 (number)|25]]''' }}, that is the only number to have an aliquot sum equal to [[6|6]],<ref name="AliSum" /> and the second of only two numbers to have a sum-of-divisors of '''31''' (the largest prime factor of 744), after {{math|1=16 = 4<sup>2</sup> }}.<Ref name="SigmaSoD" /> Specifically, the twenty-fifth prime number [[97 (number)|97]] is generated from a sum of 48 and 49. Ninety-seven is the ninth ''full repetend prime'' in decimal,<ref name="FullRepP" /> whose '''96''' repeating digits generate a sum of '''432''', equal to the abundance of 744.<ref name="AbunoN" /> 97 is also the first odd prime number that is not a [[Cluster prime|cluster prime]]<ref name="ClustP">{{Cite OEIS |A038133 |From a subtractive Goldbach conjecture: odd primes that are not cluster primes. |access-date=2023-09-26 }}</ref> (which precedes the thirty-first prime number 127 in this sequence), where the prime gap for a cluster prime is six or less;<ref>{{cite journal |last1=Blecksmith |first1=Richard |last2=Erdős |first2=Paul |last3=Selfridge |first3=J. L. |title=Cluster Primes |journal=[[The American Mathematical Monthly]] |year=1999 |volume=106 |issue=1 |pages=43 |doi=10.2307/2589585 |jstor=2589585 |zbl=0985.11041 }}</ref> the sixth and seventh primes that are not cluster primes are the forty-eighth and forty-ninth primes [[223 (number)|223]] and [[227 (number)|227]].<br />97 is the eleventh prime number {{math|1=''p'' }} of the form {{math|1=6''m'' + 1}} for a natural number  {{math|1=''m'' }}.<ref>{{Cite OEIS |A002476 |Primes of the form 6m + 1. |access-date=2023-09-26 }}</ref><br />Between the median composite totative values of 744, i.e. '''('''413, 415''')''', is 414 — the '''333'''rd composite number<ref name="CompNum" /> — with an Euler totient of '''132'''<ref name="Etot" /> that is also the prime index of 743,<ref name="PNum" /> the largest prime totative of 744. Furthermore, 414 has a sum-of-divisors of 936,<ref name="SigmaSoD" /> the '''777'''th composite number, in-turn the sum-of-divisors of 639 (palindromic with 936), itself the composite index of 777 (also palindromic). Where 414 is the 43rd number to return {{math|1=0 }} for the Mertens function, 333 is the 26th,<Ref name="Mert0" /> with 43 the fourteenth prime number.<ref name="PNum" /> Also, the twenty-fifth [[Repdigit|repdigit]] in decimal is 777.<ref name="Repd">{{Cite OEIS |A010785 |Repdigit numbers, or numbers whose digits are all equal. |access-date=2024-01-19 }}</ref><br />Where 25 is the smallest composite totative of 744, the largest such totative is 737, whose aliquot sum is [[79 (number)|79]],<ref name="AliSum" /> the twenty-second indexed prime number,<ref name="PNum" /> that is the lesser permutable prime to 97 in decimal.<ref name="PermP" /> }} Only seven numbers present in this sequence are not totatives of 744 (less-than); they are 713, 589, 527, 403, 341, 217, and 155; all of which are divisible by the eleventh prime number [[31 (number)|31]].{{efn-lr|1='''713''' is the product of the ninth and eleventh prime numbers {{math|1=23 × 31 }}, where the twenty-third prime number ([[83 (number)|83]]) is the eleventh prime of the form {{math|1=6''m'' − 1 }}; it is equal to the difference between 744 and 31, with an aliquot sum of 55<ref name="AliSum" /> and totient of 660.<ref name="Etot" /><br />'''∗''' '''589''' equal to {{math|1=19 × 31 }} is the sum of three consecutive primes (193 + 197 + 199).<ref>{{Cite OEIS |A034961 |Sums of three consecutive primes. |access-date=2023-09-24 }}</ref> It is also the ninth centered tetrahedral number, where 5 and 15 are the first two such numbers ({{math|1=121 = 11<sup>2</sup> }} is the fifth).<ref>{{Cite OEIS |A005894 |Centered tetrahedral numbers. |access-date=2023-09-24 }}</ref> It is the fourteenth third spoke of a hexagonal spiral,<ref>{{Cite OEIS |A056107 |Third spoke of a hexagonal spiral. |access-date=2023-09-24 }}</ref> and the twentieth quasi-[[Carmichael number]],<ref>{{Cite OEIS |A257750 |Quasi-Carmichael numbers. |access-date=2023-09-24 }}</ref> with a reduced totient of [[90 (number)|90]].<ref name="redTot">{{Cite OEIS |A002322 |Reduced totient function psi(n): least k such that x^k is congruent 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n. |access-date=2023-08-27 }}</ref> In the [[Spt function|spt function]], 589 is the total number of smallest parts (counted with multiplicity) in all [[Partition (number theory)|partitions]] of 15.<ref>{{Cite OEIS |A092269 |Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n. |access-date=2023-09-24 }}</ref><br />'''∗''' '''527''' equal to {{math|1=17 × 31 }} is the number of partitions of 31 with equal number of even and odd parts.<ref>{{Cite OEIS |A045931 |Number of partitions of n with equal number of even and odd parts. |access-date=2023-09-24 }}</ref> Its aliquot sum is 49<ref name="AliSum" /> whose arithmetic mean of divisors is {{math|1=12<sup>2</sup> = [[144 (number)|144]]}}, with a sum of its prime factors equal to 48. It is also the maximal number of pieces that can be obtained by cutting an [[Annulus (mathematics)|annulus]] with 31 cuts,<ref>{{Cite OEIS |A000096 |a(n) equal to n*(n+3)/2. |access-date=2023-09-24 }}</ref> and equivalent to the sum of thirty-one consecutive non-zero integers {{math|1=2 + 3 + ... + 32 }}; the second-smallest such sum after the third ''perfect number'' 496.<ref name="PerfN">{{Cite OEIS |A000396 |Perfect numbers k: k is equal to the sum of the proper divisors of k. |access-date=2023-09-24 }}</ref> }}{{efn-lr|1='''∗''' '''403''' equal to {{math|1=13 × 31 }} is the thirty-seventh number to return 0 for the [[Mertens function]],<ref name="Mert0">{{Cite OEIS |A028442 |Numbers k such that Mertens's function M(k) (A002321) is zero. |access-date=2023-09-24 }}</ref> and the value of the sum-of-divisors of '''144''', the first number to generate this value.<ref name="SigmaSoD" /> It is also the first number to hold a totient of [[360 (number)|360]], of twenty-five such numbers; with '''549''' the sixth (and the 447th composite number, a value that is itself the 360th composite)<ref name="CompNum" /> and {{math|1=693 = 144 + 549 }} the tenth.<ref name="Etot" /> Also, 403 is the 284th ''[[Arithmetic number|arithmetic number]]'', since its four divisors hold an arithmetic mean of {{math|1='''[[112 (number)|112]]''' = 7 × 4 × 4 }}, the second such number to hold this value.<ref name="ArithNum0" /><ref name="ArithNum" /><hr>The 112nd prime number is [[613 (number)|613]],<ref name="PNum" /> which is the sum of successive composite indexes of 382 and 384 (respectively, 306 and 307),<ref name="CompNum" /> where '''383''' is the '''28'''th full repetend prime number in decimal; '''613''' is also equal to the sum of prime indices of '''''seven consecutive''''' full repretend prime numbers (in decimal): 389, 419, 433, 461, 487, 491, 499, respectively with prime indices 77, 81, 84, 89, 93, 94, and 95<ref name="PNum "/> (and between the 29th and 35th such primes)<ref name="FullRepP" /> — all following 383, where the next full repetend prime after 499 is the 96th prime number 503.<ref name="PNum" /> 383 is also the second number, after 19, to generate a full [[Prime reciprocal magic square|prime reciprocal magic square]] in decimal whose rows and diagonals generate the same [[Magic constant|magic constant]] (from its repeating digits) of '''1719'''.<ref name="WSymes" /> Within the list of prime numbers, '''2017''' and '''2027''' are respectively the 306th and 307th,<ref name="PNum" /> where the 1719th composite number is 2026.<ref name="CompNum" />{{Bullet list |The arithmetic mean of divisors of '''2022''', which lies midway between the prime gap of ''eleven'' integers (inclusive) just noted, is 507, equal to the sum of ''prime factors'' of {{math|1='''2012''' = 2<sup>2</sup> × 503 }} (inclusive of multiplicities), whose arithmetic mean of divisors in-turn is '''588''' (half 1176, the aliquot sum of 744) and the second such number to have this value (588), after 1304.<ref name="ArithNum0" /><ref name="ArithNum" /> |1685, on the other hand, is the third number to have an average of divisors that is 507 after 2022, and whose sum-of-divisors (of 1685) is '''2028''',<ref name="SigmaSoD" /> a number lying between the 307th and 308th prime numbers, 2027 and 2029, the '''62'''nd pair of twin primes);<ref name="PNum" /><ref name="TwinPs" /> 2011 is prime as well, the 305th prime number, with 305 having a composite index '''242'''.<ref name="PNum" /><ref name="CompNum" />|1715, the composite index of 2022,<ref name="CompNum" /> is the fourth of thirteen numbers to hold a totient of {{math|1=1176 = '''''[[Aliquot sum|s]]'''''(744) }},<ref name="Etot" /> with an arithmetic mean of divisors equal to [[300 (number)|300]], which is the twenty-fourth triangular number.<ref name="TriangNum" /> |'''2023''', midway between primes 2017 and 2029, has an aliquot sum of 433 (or 432, the ''abundance'' of 744, when not counting 1),<ref name="AliSum" /> and a Euler totient of {{math|1=1632 = 1176 + 456 }} (the aliquot sum of 744, plus the only number to hold an aliquot sum of 744).<ref name="Etot" /> }} Where 383 repeats 382 digits in base ten, 382 is an arithmetic number too, specifically the first number to have an average of divisors of 144<ref name="ArithNum0" /><ref name="ArithNum" /> while being the seventh number to hold a sum-of-divisors of {{math|1='''576''' = 24<sup>2</sup> }}.<ref name="SigmaSoD" /> Also, {{math|1=613 + 744 = 1357 }}, alongside {{math|1=744 − 613 = 131 }} (both with successive differences of 2 between digits); the latter is the twelfth full repetend prime number in decimal that repeats 130 digits with a sum of {{math|1='''585''' = 8<sup>0</sup> + 8<sup>1</sup> + 8<sup>2</sup> + 8<sup>3</sup> }}, where the '''48'''th full repetend prime, on the other hand, is 743, the largest prime totative of 744 (or '''49'''th in the near-identical sequence of ''long primes'', that instead begins with 2, whose decimal expansion of {{math|1={{sfrac|1|''p'' }} }} has period {{math|1=''p'' − 1 }}).<ref name="FullRepP" /><ref name="LongPP" /> <hr>Meanwhile, 112 (the 82nd composite) is the composite index of {{math|1=147 = 48 + 49 + 50 }}<ref name="CompNum" /> with an aliquot sum of {{math|1=81 = 9<sup>2</sup> }}<ref name="AliSum" /> and average of divisors of [[38 (number)|38]].<ref name="ArithNum0" /><ref name="ArithNum" /> [[147 (number)|147]] itself is the composite index of the ''nineteenth'' triangular number, [[190 (number)|190]],<ref name="TriangNum" /> equal to the totient and reduced totient of 382.<ref name="Etot" /><ref name="redTot" /> {{math|1=744 − 383 = 361 = 19<sup>2</sup> }}, where 383 is the {{math|1=76 = 2<sup>2</sup> × 19 }} indexed prime number<ref name="PNum" /> (and [[304 (number)|304]] fourfold seventy-six, the prime index of '''2003''', which is the smallest prime number greater than 2000).<ref name="PNum" /><br />'''∗''' '''341''' equal to {{math|1=11 × 31 }} is the sum between seven consecutive primes {{math |1=(37 + 41 + 43 + 47 + 53 + 59 + 61) }}, of prime indexes 12 through [[18 (number)|18]]. It is the smallest [[Fermat pseudoprime]] to [[Binary number|binary]],<ref>{{Cite OEIS |A001567 |Fermat pseudoprimes to base 2, also called Sarrus numbers or Poulet numbers. |access-date=2023-09-28 }}</ref> and the sixth centered cube number.<ref>{{Cite OEIS |A005898 |Centered cube numbers: n^3 + (n+1)^3. |access-date=2023-09-28 }}</ref> It is the number of partitions of 31 such that every part occurs with the same multiplicity,<ref>{{Cite OEIS |A047966 |a(n) equal to Sum_{ d divides n } q(d), where q(d) is A000009 equal to number of partitions of d into distinct parts. |access-date=2023-09-28 }}</ref> and the thirty-first number in the [[Moser–de Bruijn sequence]] to be a sum of distinct powers of '''[[4|4]]''' ({{math|1=4<sup>4</sup> + 4<sup>3</sup> + 4<sup>2</sup> + 4<sup>1</sup> + 4<sup>0</sup> }}).<ref>{{Cite OEIS |A000695 |Moser-de Bruijn sequence: sums of distinct powers of 4. |access-date=2023-09-28 }}</ref><ref>{{Cite OEIS |A053699 |a(n) is n^4 + n^3 + n^2 + n + 1. |access-date=2023-09-28 }}</ref> 341 is also the eleventh octagonal number,<ref>{{Cite OEIS |A000567 |Octagonal numbers: n*(3*n-2). Also called star numbers. |access-date=2023-09-28 }}</ref> equal to the number of [[Vertex (geometry)|nodes]] in a regular eleven-sided undecagon with all diagonals drawn.<ref>{{Cite OEIS |A007569 |Number of nodes in regular n-gon with all diagonals drawn. |access-date=2023-09-28 }}</ref> {{math|1='''s'''(341) = 43 }},<ref name="AliSum" /> with an arithmetic mean of divisors equal to [[96 (number)|96]].<ref name="ArithNum" /> It is also the tenth [[Jacobsthal number]],<ref>{{Cite OEIS |A001045 |Jacobsthal sequence (or Jacobsthal numbers)... |access-date=2023-09-28 }}</ref> and the total number of largest parts (or equivalently sum of smallest parts) in all partitions of [[16 (number)|16]].<ref>{{Cite OEIS |A046746 |Sum of smallest parts of all partitions of n. |access-date=2023-09-28 }}</ref> }}{{efn-lr|1='''∗''' '''[[217 (number)|217]]''' equal to {{math|1=7 × 31 }} is the ninth [[Centered hexagonal number|centered hexagonal number]], which also includes as previous members {7, '''19''', 37, 61, 91, '''127''', 169}<ref>{{Cite OEIS |A003215 |Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice). |access-date=2023-09-24 }}</ref> where [[127 (number)|127]] is the sixth member that is also the thirty-first prime number. 217 is also the sixth non-trivial dodecagonal number<ref>{{Cite OEIS |A051624 |12-gonal (or dodecagonal) numbers: a(n) equal to n*(5*n-4). |access-date=2023-09-24 }}</ref> and third non-trivial centered 36-gonal number.<ref>{{Cite OEIS |A195316 |Centered 36-gonal numbers. |access-date=2023-09-24 }}</ref> It is the third Fermat pseudoprime to [[Quinary|quinary]] after 124 (a divisor of 744) and 4,<ref>{{Cite OEIS |A005936 |Pseudoprimes to base 5. |access-date=2023-09-24 }}</ref> and the fifteenth [[Blum integer]] whose distinct prime factors are congruent to {{math|1=3 mod 4 }}.<ref>{{Cite OEIS |A016105 |Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4). |access-date=2023-09-24 }}</ref><br />'''✶''' '''[[155 (number)|155]]''' equal to {{math|1=5 × 31 }} has an aliquot sum of [[37 (number)|37]]<ref name="AliSum" /> and an arithmetic mean of divisors equal to 48.<ref name="Etot" /> It is the third number equal to the sum from its lowest prime factor through its largest, the only smaller numbers with the same property are [[10|10]] and [[39 (number)|39]]<ref>{{Cite OEIS |A055233 |Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor. |access-date=2023-09-24 }}</ref> which generate a sum of {{math|1=49 = 7<sup>2</sup> }}; the next such number is 371, 1 less than half of 744. There are one hundred and fifty-five total [[Primitive permutation group|primitive permutation group]]s {{math|1=('''G, X''') }} of degree {{math|1='''81''' }} that only preserve partitions on the set {{math|1='''X''' }} by {{math|1='''G''' }} that are trivial;<ref>{{Cite OEIS |A000019 |Number of primitive permutation groups of degree n. |access-date=2023-09-24 }}</ref> the sum of the number of all permutation groups of lower degrees is [[666 (number)|666]], which is [[Doubly triangular number|doubly triangular]],<ref>{{Cite OEIS |A002817 |Doubly triangular numbers: a(n) equal to n*(n+1)*(n^2+n+2)/8. |access-date=2023-09-24 }}</ref> since it represents the sum of the first thirty-six integers, where [[36 (number)|36]] is itself the eighth triangular number.<ref name="TriangNum" /> The '''148''' repeating digits of the thirteenth full repetend prime number in decimal [[149 (number)|149]] (that lies between 109 and 181 in this sequence) <ref name="FullRepP" /> sum to 666.<br />The one hundred and fifty-fifth indexed ''[[Practical number|practical number]]'' is 744,<ref name="PractNum" /> where the 155th arithmetic number is the forty-eighth prime number [[223 (number)|223]]: the first number to hold an arithmetic mean of divisors of 112, preceding '''403'''.<ref name="ArithNum0" /><ref name="ArithNum" />}}
The remaining 130 totatives are [[1|1]] and all the primes between [[5|5]] and 743 except for 31 (all prime numbers less than 744 that are not part of its prime factorization) where its largest prime totative of [[743 (number)|743]]{{efn-lr|1='''743''' is the twenty-sixth or twenty-seventh '''''[[Lucky numbers of Euler|lucky number of Euler]]''''' of the form {{math|1=''n''<sup>2</sup> + ''n'' + 41 }} to yield consecutive prime numbers.<ref>{{Cite OEIS |A202018 |a(n) equal to n^2 + n + 41. |access-date=2023-12-15 }}</ref><ref>{{Cite OEIS |A005846 |Primes of the form n^2 + n + 41. |access-date=2023-12-15 }}</ref> It is also the twenty-sixth indexed beginning of a ''maximal chain of primes'' of the form {{math|1=''p''(''k''), ..., ''p''(''k''+''r'') }} for {{math|1=''r'' ≥ 1 }}.<ref>{{Cite OEIS |A152658 |Beginnings of maximal chains of primes. |access-date=2023-12-15 }}</ref> It is the forty-eighth full repetend prime in decimal with [[Primitive root modulo n|primitive root 10]],<ref name="FullRepP" /> or similarly the forty-ninth ''long-period prime'' when including [[2|2]] in this sequence.<ref name="LongPP">{{Cite OEIS |A006883 |Long period primes: the decimal expansion of 1/p has period p-1.|access-date=2023-12-15 }}</ref> It is the forty-eighth member in base ten such that every suffix is prime, in which repeatedly removing the most significant digit yields a prime number at every step, until a single-digit prime number remains.<ref>{{Cite OEIS |A033664 |Every suffix is prime. |access-date=2023-12-15 }}</ref><br /> Specifically, 743 is the total sum of odd parts in all partitions of 13,<ref>{{Cite OEIS |A066967 |Total sum of odd parts in all partitions of n. |access-date=2023-12-15 }}</ref> and the number of partitions of 31 into partition numbers.<ref>{{Cite OEIS |A007279 |Number of partitions of n into partition numbers. |access-date=2023-12-15 }}</ref> Particularly, it is the number of ways to partition 7 labeled elements into sets of sizes of at least 2, and order the sets,<ref>{{Cite OEIS |A032032 |Number of ways to partition n labeled elements into sets of sizes of at least 2 and order the sets. |access-date=2023-12-15 }}</ref> and furthermore, the number of [[Partition function (number theory)#Strict partition function|strict integer partitions]] of [[37 (number)|37]] with no part divisible by any and all the other parts.<ref>{{Cite OEIS |A343377 |Number of strict integer partitions of n with no part divisible by all the others. |access-date=2023-12-15 }}</ref><br />Where 743 is the largest prime totative of 744, it represents ''the sum of all parts of all partitions of all positive integers'' less than or equal to '''25''' into an ''odd number of equal parts'' or, equivalently, '''''consecutive parts''''';<ref>{{Cite OEIS |A285900 |Sum of all parts of all partitions of all positive integers less than or equal to n into consecutive parts. |access-date=2023-12-15 }}</ref> with twenty-five the smallest composite totative of 744. }} has a prime index of [[132 (number)|132]] (the smallest [[Digit-reassembly number|digit-reassembly number]] in decimal).<ref name="Osiris">{{Cite OEIS |A319274 |Osiris or Digit re-assembly numbers: numbers that are equal to the sum of permutations of subsamples of their own digits. |access-date=2023-09-02 }}</ref>{{efn-lr|1=These prime totatives generate a sum of 44,647 (a prime count, the 4641th prime number, an index whose arithmetic mean of divisors is 504)<ref name="ArithNum" /> while the composite totatives collectively generate a sum of 44,632 (one less than a prime number, the 4639th prime and 626th ''super-prime'',<ref name="SupP" /> the latter an index whose value is the fourth and largest number to have an aliquot sum of 316);<ref name="AliSum" /> these sums have a difference of [[15 (number)|15]], or otherwise collectively have a range of 16 numbers between them, inclusive.<br />On the other hand, differences and sums between the six numbers congruent {{math|1=±1 mod 6 }} that are not totatives of 744 are themselves equivalent to divisors of 744, since they are biprimes in proportion with 31 (for example {{math|1=713 − 589 = 124 }} and {{math|1=713 − 527 = 186 }}), while also generating other relevant equalities such as {{math|1=713 − 217 = 496 }}.<br />The greatest difference between the largest (713) and smallest (155) of these totatives is 558, the seventh number such that the sum of largest prime factors of numbers from 1 to 558 is divisible by 558, the previous and sixth indexed number is '''[[62 (number)|62]]''',<ref>{{Cite OEIS |A088825 |Numbers n such that the sum of largest prime factors of numbers from 1 to n is divisible by n. |access-date=2023-09-25 }}</ref> the tenth-largest divisor of 744, whose sum-of-divisors is equal to 96;<ref name="SigmaSoD" /> the fifth number in this sequence is [[32 (number)|32]] (which divides 96).<br /> The difference between 558 and 62 is also 496, equal to {{math|1=713 − 155 − 62 }}; and importantly, 558 is twice 279, which is the ones' complement of 744 in binary in its decimal representation. }} On the other hand, only three numbers hold a totient of 744; they are [[1000 (number)#1100 to 1199|1119]], 1492, and 2238.<ref name="Etot" />{{efn-lr|1=The difference between 2238 and 1492 is 746, where 2238 is twice 1119. Specifically, {{math|1=746 = 2 × 373 = 1<sup>5</sup> + 2<sup>4</sup> + 3<sup>6</sup> = 2! + 4! + 6! }} is [[Nontotient|nontotient]],<ref name="nontot">{{Cite OEIS |A005277 |Nontotients: even numbers k such that phi(m) equal to k has no solution. |access-date=2023-08-27 }}</ref> and equal to the number of non-normal [[Magic square|''orthomagic squares'']] with magic constant of 6.<ref>{{Cite OEIS |A321719 |Number of non-normal semi-magic squares with sum of entries equal to n. |access-date=2023-08-27 }}</ref><br />Its totient value of 372 is half of 744,<ref name="Etot" /> that is also 1 less than 373, the '''74'''th indexed prime number<ref name="PNum" /> that is the smallest difference between numbers that have totients equal to 744 (i.e., {{math|1=1119 − 1492 }}). 373 is the eleventh ''two-sided'' left-and-right [[Truncatable prime|truncatable prime]] number in decimal, of a total fifteen such primes in base ten.<ref>{{Cite OEIS |A020994 |Primes that are both left-truncatable and right-truncatable. |access-date=2023-12-13 }}.</ref> While '''373''' is the smallest difference between numbers that have a totient value of 744, its digit representation in decimal is the mirror permutation of the digits of '''737''', the largest composite totative of 744.<br />The sum of these three numbers whose totients equal 744 is {{math|1=1119 + 1492 + 2238 = 4849 }}. }}

744 is the sixth number <math>n</math> whose totient value has a sum-of-divisors equal to <math>n</math>: <math>\sigma(\varphi(744))=744</math>.<ref name="SigmPhi">{{Cite OEIS |A018784 |Numbers n such that sigma(phi(n)) is n. |access-date=2023-07-16 }}</ref> Otherwise, the [[Aliquot sum|aliquot sum]] of 744, which represents the sum of all divisors of 744 aside from itself, is 1176<ref name="AliSum" /> which is the forty-eighth [[Triangular number|triangular number]],<ref name="TriangNum" /> and the [[Binomial coefficient|binomial coefficient]] <math>\operatorname {C(49,2)}</math> present inside the forty-ninth row of [[Pascal's triangle]].<ref>{{Cite OEIS |A017765 |Binomial coefficients C(49,n). |access-date=2023-09-02 }}</ref>{{efn-lr|1='''1176''' is also one of two middle terms in the twelfth row of a <math>\operatorname {(2,3)}-</math>Pascal triangle.<ref>{{Cite OEIS |A029600 |Numbers in the (2,3)-Pascal triangle (by row). |access-date=2023-09-02 }}<br />The row is {2, 23, 120, 375, 780, 1134, '''1176''', 870, 450, 155, 32, 3}.</ref> In the triangle of [[Narayana number]]s, 1176 appears as the fortieth and forty-second terms in the eighth row,<ref>{{Cite OEIS |A001263 |Triangle of Narayana numbers T(n,k) is C(n-1,k-1)*C(n,k-1)/k with 1 less than or equal to k less than or equal to n, read by rows. Also called the Catalan triangle. |access-date=2023-09-02 }}</ref> which also includes 336 (the totient of 1176)<ref name="Etot" /> and [[36 (number)|36]] (the [[Square number|square]] of 6). Inside the triangle of [[Lah number]]s of the form <math>\textstyle {L(n,k) = \left\lfloor {n \atop k} \right\rfloor}</math>, 1176 is a member with {{math|1=''n'' = 8 }} and {{math|1=''k'' = 6}}.<ref>{{Cite OEIS |A105278 |Triangle read by rows: T(n,k) equal to binomial(n,k)*(n-1)!/(k-1)!. |access-date=2023-09-02 }}</ref> It is a self-[[Fibonacci number]]; the fifty-first indexed member where in its case <math>F_{1176}</math> divides <math>F_{F_{1176}}</math>,<ref>{{Cite OEIS |A127787 |Numbers n such that F(n) divides F(F(n)), where F(n) is a Fibonacci number. |access-date=2023-09-02 }}</ref> and the forty-first '''6'''-[[Almost prime|almost prime]] that is divisible by exactly six primes with [[Multiplicity (mathematics)|multiplicity]].<ref>{{Cite OEIS |A046306 |Numbers that are divisible by exactly 6 primes with multiplicity. |access-date=2023-09-02 }}</ref> }} In total, only seven numbers have sums of divisors equal to 744; they are: '''240''',{{efn-lr|1='''240''' is the thirty-first ''quarter square'', where 49 is the 14th.<ref name="QuartSq" /> Relatedly:<ref name="QuartSq">{{Cite OEIS |A002620 |Quarter-squares: a(n) equal to floor(n/2)*ceiling(n/2). Equivalently, a(n) is floor(n^2/4). |access-date=2024-01-30 }}</ref>{{Bullet list|The maximum product of two integers whose sum is 31 is 240. |240 is the sum of the positive integers less than 31 that have the opposite [[Parity (mathematics)|parity]] as 31. |240 is the number of nonempty subsets of {{math|1={1, 2, ..., 31} }} that contain exactly one odd and one even number. |There are 240 arithmetic progressions of three terms by any means which can be extracted from the set of the first thirty-one natural numbers (starting from 1). }} From the starting position in standard chess, 240 is the minimum number of captures by pawns of the same color to place 31 of them on the same file (column). 240 is also the maximal number of edges that a triangle-free graph of 31 vertices can have.<ref name="QuartSq" /> }} 350, 366, 368, 575, 671, and 743.<ref name="SigmaSoD">{{Cite OEIS |A000203 |a(n) equal to sigma(n), the sum of the divisors of n. Also called sigma_1(n). |access-date=2023-09-11 }}</ref>{{efn-lr|1=The sum of all these seven integers whose sum-of-divisors are in equivalence with 744 is '''3313''', the 466th prime number<ref name="PNum" /> and 31st [[Balanced prime|balanced prime]],<ref>{{Cite OEIS |A006562 |Balanced primes (of order one): primes which are the average of the previous prime and the following prime. |access-date=2024-01-06 }}</ref> as the middle member of the 49th triplet of [[Sexy prime|sexy prime]]s {{math|1='''('''3307, 3313, 3319''')''' }};<ref>{{Cite OEIS |A046119 |Middle member of a sexy prime triple: value of p+6 such that p, p+6 and p+12 are all prime, but p+18 is not (although p-6 might be). |access-date=2024-01-06 }}</ref> it is respectively the '''178'''th and 179th such prime {{math|1=''p'' }} where {{math|1=''p'' − 6 }}<ref>{{Cite OEIS |A046117 |Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.) |access-date=2024-01-06 }}</ref> and {{math|1=''p'' + 6 }}<ref>{{Cite OEIS |A023201 |Primes p such that p + 6 is also prime (Lesser of a pair of sexy primes.) |access-date=2024-01-06 }}</ref> are prime — where 178 is the 132nd composite number,<ref name="CompNum" /> itself the prime index of the largest number (743) to hold a sum-of-divisors of 744.<ref name="PNum" /><ref name="SigmaSoD" /> 3313 is also the 24th centered dodecagonal or ''star'' number<ref name="StarN">{{Cite OEIS |A003154 |Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1. |access-date=2024-01-06 }}</ref> (and 15th that is prime).<ref>{{Cite OEIS |A083577 |Prime star numbers. |access-date=2024-01-06 }}</ref> Divided into two numbers, 3313 is the sum of 1656 and 1657, the latter being the '''260'''th prime number, an index value in-turn that is the first of five numbers to have a sum-of-divisors of '''588''',<ref name="SigmaSoD" /> which is ''half'' of 1176 (the aliquot sum of 744);<ref name="AliSum" /> its arithmetic mean of divisors (of 260), on the other hand, is equal to '''49'''.<ref name="ArithNum0" /><ref name="ArithNum" /> Furthermore, 260 is the average of divisors of '''2232''' (the fourth largest of five numbers to hold this value), which is thrice 744. }} When only the fourteen [[Divisor#Further notions and facts|proper divisor]]s of 744 are considered, then the sum generated by these is [[1000 (number)#1100 to 1199|1175]], whose six divisors contain an arithmetic mean of 248,<ref name="ArithNum" />{{efn-lr|1=[[248 (number)|248]] is twice 124 and half [[496 (number)|496]]; the latter which is the third [[Perfect number|perfect number]]<ref name="PerfN" /> — like 6 and 28 (twice 14) — that is also the thirty-first [[Triangular number|triangular number]],<ref name="TriangNum" /> with form {{math|1=2<sup>''p'' - 1</sup>(2<sup>''p''</sup> − 1) }} and {{math|1=''p'' = 5 }} by the Euclid-Euler theorem. The totient of 248 is 120,<ref name="Etot" /> with eight divisors that produce an integer arithmetic mean of 60;<ref name="ArithNum" /> 496 has a totient of 240 like 744.<br />Where the sum between 744 and the totient {{math|1=256 = 2<sup>8</sup> }} of 960 (the ''Zumkeller half'' of 744) is equal to {{math|1=1000 = 10<sup>3</sup> }}, their difference is equivalent with {{math|1=488 = 240 + 248. }}<br />'''248''' is also the sum of prime indices of ''seven consecutive'' full repetend prime numbers in decimal: '''109''', 113, 131, 149, 167, 173, '''181''',<ref name="FullRepP" /> respectively with 29, 30, 32, 35, 39, 40, and 42 prime indices;<ref name="PNum" /> where 248 is the one hundred and ninety-fourth composite number, and 194 the '''149'''th composite number,<Ref name="CompNum" /> the middle term in this consecutive sequence of seven full repetend primes. 149 itself is a strong concatenation of {{math|1=14 = 7 + 7 }} and {{math|1=49 = 7 × 7 }}, joined in the middle by '''4''', where '''7''' is the fourth prime number. Note, too, that in the list of successive prime and composite indices of 181, starting with its prime index of [[42 (number)|42]], the sum generated is {{math|1= 109 = 42 + 28 + 18 + 10 + 5 + 3 + 2 + 1 + 0 }}; with 109 the composite index of '''144'''.<ref name="PNum" /><ref name="CompNum" /> }} the third (or fourteenth) largest [[Divisor|divisor]] of 744. Only one number has an aliquot sum that is 744, it is 456.<ref name="AliSum" />{{efn-lr|1='''456''' is the sixth icosahedral number,<ref>{{Cite OEIS |A006564 |Icosahedral numbers: a(n) equal to n*(5*n^2 - 5*n + 2)/2. |access-date=2023-09-23 }}</ref> one less than the '''88'''th prime number 457 (itself an index that represents the '''64'''th composite).<ref name="PNum" /><ref name="CompNum" /> The list of icosahedral numbers includes [[124 (number)|124]] as the fourth indexed member, where this number is also the third non-unitary '''stella octangula number''' after [[51 (number)|51]] and [[14 (number)|14]],<ref name="StllOct">{{Cite OEIS |A007588 |Stella octangula numbers: a(n) equal to n*(2*n^2 - 1). |access-date=2024-01-17 }}</ref> and the ''seventh'' [[Untouchable number|untouchable number]], after 120 and 96;<ref>{{Cite OEIS |A005114 |Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065) |access-date=2023-12-17 }}</ref> it is the eleventh non-unitary divisor of 744, equal to the sum of all the prime numbers that are not part of its prime factorization, less than 31 (i.e. {{math|1=5 + 7 + ... + 29}}). It is furthermore the fourth ''perfectly partitioned'' number after 1, 2 and 3.<ref>{{Cite OEIS |A051177 |Perfectly partitioned numbers: numbers k that divide the number of partitions p(k). |access-date=2023-09-23 }}</ref> Where 1176 and 456 collectively have a ''range'' of seven hundred and twenty-one integers, the difference between 721 and 1000 is '''279''' (the decimal equivalent of the ones' complement of 744 in binary); where also, {{math|1='''1455''' = 1176 + 279 }} holds an arithmetic mean of divisors that is 294,<ref name="ArithNum0" /><ref name="ArithNum" /> which is one-fourth 1176. This sum also represents the number of [[Permutation group|permutation group]]s of degree {{math|1=6 }},<ref>{{Cite OEIS |A005432 |Number of permutation groups of degree n (or, number of distinct subgroups of symmetric group S_n, counting conjugates as distinct). |access-date=2023-12-30 }}</ref> and the number of partitions of '''48''' into distinct parts such that the number of parts is odd,<ref>{{Cite OEIS |A067659 |Number of partitions of n into distinct parts such that number of parts is odd. |access-date=2023-12-30 }}</ref> and the number of partitions of 48 into distinct parts such that the number of parts is even.<ref>{{Cite OEIS |A067661 |Number of partitions of n into distinct parts such that number of parts is even. |access-date=2023-12-30 }}</ref> 456 is also, specifically, the number of partitions of the twenty-ninth composite number '''[[44 (number)|44]]''' into prime parts (where on the other hand, 744 the number of partitions of 49 into prime parts).<ref name="PartNPP" /><br />Furthermore, regarding twenty-nine:{{Bullet list |the sum of the first ten prime numbers up to 29 yields [[129 (number)|129]],<ref>{{Cite OEIS |A127337 |Numbers that are the sum of 10 consecutive primes. |access-date=2023-12-16 }}</ref> which is the number of prime totatives of 744 (with the ''fifth'' such sum of ten consecutive prime numbers being 264, or twice 132). 129 is also the 97th indexed composite number,<ref name="CompNum" /> where 97 is the 25th prime number,<ref name="PNum" /> |the twenty-ninth prime number, on the other hand, is [[109 (number)|109]],<ref name="PNum" /> which is the number of composite totatives of 744 greater than {{math|1=25 = 5<sup>2</sup> }}, its smallest composite totative; |the twenty-ninth record for the sum-of-divisors of a number {{math|1=''n'' }} is set by {{math|1='''σ'''(744) = 1920 }}.<ref name="RecSigm" /> }} The count of prime and composite totatives of 744, ''sans'' including 1, is '''[[239 (number)|239]]''', which is the '''52'''nd indexed prime number.<ref name="PNum" /> It is also the sum of prime pairs (23, 29) and (89, 97) that respectively generate consecutive record prime gaps of '''6''' and '''8''', which precede the record gap of '''14''' (the sum of the previous two, which also multiply to '''48''') between the thirtieth and thirty-first prime numbers (113, 127),<ref name="RecPG" /><ref name="IoPG" /><ref name="PG" /><ref name="IoLPGP" /><ref name="IoGPGP" /> which sum to '''240''', the Euler totient of 744.<ref name="Etot" /><br />Of highlight, 239 is the only number in concomitance to require a maximum number of [[Square number|squares]] (four), a maximum number of cubes (nine), and a maximum number of [[Fourth power|fourth power]]s (nineteen) to express it<ref name="HAKM">{{Cite report |last1=Beeler |first1=M. |last2=Gosper |first2=R.W. |last3=Schroeppel |first3=R. |date=1972-02-29 |title=HAKMEM |url=https://w3.pppl.gov/~hammett/work/2009/AIM-239-ocr.pdf |publisher=[[Organization:MIT Computer Science and Artificial Intelligence Laboratory|MIT Computer Science and Artificial Intelligence Laboratory]] |page=24 (ITEM 63) |docket=[[HAKMEM|AIM 239]] |access-date=2023-12-19 |via=[[Organization:Princeton Plasma Physics Laboratory|Princeton Plasma Physics Laboratory]] }}<br />
:"The joys of 239 are as follows:
:pi = 16 arctan (1/5) − 4 arctan(1/239),
:which is related to the fact that 2 * 13<sup>4</sup> − 1 {{=}} 239<sup>2</sup>,
:which is why 239/169 is an approximant (the 7th) of √2.
:arctan(1/239) {{=}} arctan(1/70) − arctan(1/99)
:{{nbsp|23}}{{=}} arctan(1/408) + arctan(1/577)
:239 needs 4 squares (the maximum) to express it.
:239 needs 9 cubes (the maximum, shared only with 23) to express it.
:239 needs 19 fourth powers (the maximum) to express it.
:(Although 239 doesn't need the maximum number of fifth powers.)
:1/239 {{=}} .00418410041841..., which is related to the fact that
:1,111,111 {{=}} 239 * 4,649.
:The 239th Mersenne number, 2<sup>239</sup> − 1, is known composite,
:{{nbsp|10}}but no factors are known.
:239 {{=}} 11101111 base 2.
:239 {{=}} 22212 base 3.
:239 {{=}} 3233 base 4.
:There are 239 primes < 1500.
:[https://www.youtube.com/watch?v=ChSdO6BK6eo K239] is Mozart's only work for 2 orchestras.
:Guess what memo this is.
:And 239 is prime, of course."</ref> — the only other number to require nine cubes is the ''ninth'' prime number [[23 (number)|23]], which is the index of 744 in the list of thirty-one integers to hold a Euler totient value of 240;<ref name="Etot" /> 23 and 239 are also the third and fourth numbers {{math|1=''n'' }} (after 1, 2, and 5) that represent the number of partitions of {{math|1=3<sup>''n''</sup> }}, here {{math|1=3<sup>3</sup> }} and {{math|1=3<sup>4</sup> }} respectively, into powers of '''[[3|3]]'''.<ref>{{Cite OEIS |A078125 |Number of partitions of 3^n into powers of 3 }}</ref> When 239 is multiplied with the 628th prime number (4649),<ref name="PNum" /> it yields the seventh [[Repunit|repunit]] in decimal ('''1111111'''),<ref name="HAKM" /> a number whose arithmetic mean of divisors is {{math|1=279000 = 279 × [[1000 (number)|1000]] }}.<ref name="ArithNum0" /><ref name="ArithNum" /><br /> While the only two positive square stella octangula numbers are {{math|1=1 }} and {{math|1=9653449 = 3107<sup>2</sup> = (13 × 239)<sup>2</sup>}},<ref>{{Cite thesis |first=Samir |last=Siksek |title=Descents on Curves of Genus I |url=https://ore.exeter.ac.uk/repository/bitstream/handle/10871/8323/SisekS_fm.pdf?sequence=1 |type=Ph.D. thesis |publisher=[[Organization:University of Exeter|University of Exeter]] |location=Exeter, UK |year=1995 |pages=16–17 |hdl=10871/8323 |s2cid=118846642 }}</ref> — of indexes {{math|1=1=1<sup>2</sup> }} and {{math|1=169 = 13<sup>2</sup> }} in this sequence<ref name="StllOct" /> — the only solutions to {{math|1=''y''<sup>2</sup> + 1 = 2''x''<sup>4</sup> }} in positive integers are {{math|1=(''x'', ''y'') = (1, 1) }} and {{math|1=(13, 239) }}.<ref>{{Cite OEIS |A229384 |Positive integer solutions y1, x1, y2, x2 to Ljunggren's equation x^2 + 1 equal to 2y^4. |access-date=2024-01-19 }}</ref> The [[Fraction|fraction]] {{math|1={{sfrac|239|(13<sup>2</sup>)}} }} is the seventh approximation of the [[Convergent series|converging]] [[Continued fraction|continued fraction]] for {{math|1=√2 }},<ref>{{Cite OEIS |A001333 |Numerators of continued fraction convergents to sqrt(2). |access-date=2023-12-19 }}</ref> where {{math|1=16{{hair space}}arctan{{hair space}}({{sfrac|1|5}}) − 4{{hair space}}arctan{{hair space}}({{sfrac|1|239}}) }} is an approximation for '''[[Pi|{{large|{{pi}}}}]]''' [[Radian|radian]]s.<ref name="HAKM" /><hr>{{math|1=5 × 239 = 1195 }}, the '''fourth''' number to have an [[Arithmetic mean|arithmetic mean]] of [[Divisor|divisor]]s equal to '''[[360 (number)|360]]''',<ref name="ArithNum0" /><ref name="ArithNum" /> where specifically, {{math|1=[[52 (number)|52]] = [[13 (number)|13]] × 4 = [[26 (number)|26]] × 2 }}, the prime index of two hundred and thirty-nine, is the fifth [[Bell number]] that counts the number of ways to partition a set of ''five'' labeled elements.<ref>{{Cite OEIS |A000110 |Bell or exponential numbers: number of ways to partition a set of n labeled elements. |access-date=2023-12-19 }}</ref> Three-hundred and sixty, the smallest number with twenty-four divisors,<ref name="LeastD" /> is the sixth of only seven ([[Highly composite number|highly composite]]) numbers such that no number less than twice as much has more divisors.<ref>{{Cite OEIS |A072938 |Highly composite numbers that are half of the next highly composite number. |access-date=2023-12-19 }}</ref> It is also divisible by all integers less than or equal to ten, except for seven, itself a sum of integers that yields '''48'''. }}

=== Graph theory ===

The number of [[Eulerian path#Definition|Euler tours]] (or '''Eulerian cycles''') of the [[Complete graph|complete]], ''undirected'' graph <math>K_{6}</math> on six [[Vertex (graph theory)|vertices]] and fifteen [[Edge (geometry)|edges]] is 744.<ref>{{Cite OEIS |A350028 |Number of Euler tours of the complete graph on n vertices (minus a matching if n is even). |access-date=2023-07-16 }}</ref> On seven vertices, there are 129,976,320 Euler tours. These can only be generated on complete graphs with at least three vertices; the number of tours for three, four and five vertices are respectively 2, 2 and [[264 (number)|264]] (the latter is the second digit-reassembly number in base ten).<ref name="Osiris" /> On the other hand, the number of Euler tours of the complete [[Directed graph|digraph]], or ''directed graph'', on four vertices is [[256 (number)|256]], while on five vertices it is 972,000 (and 247,669,456,896 on six vertices), by the [[BEST theorem]].<ref>{{Cite OEIS |A232545 |Number of Euler tours of the complete digraph on n vertices. }}</ref>

Regarding the largest prime totative of 744, there are (aside from the sets that are the union of all such solutions), {{Bullet list
|743 [[Independent set (graph theory)|independent vertex sets]] in the 4 by 4 [[Toroidal graph|torus]] grid graph,<ref>{{Cite OEIS |A027683 |Number of independent vertex sets of the n X n torus grid graph. |access-date=2023-12-15 }}</ref>
|743 independent vertex sets in a four-dimensional 16-vertex [[Hypercube graph|hypercube graph]],<ref>{{Cite OEIS |A027624 |Number of independent vertex sets in the n-hypercube graph Q_n. |access-date=2023-12-15 }}</ref> and
|743 connected [[Cubic graph|cubic graph]]s with 16 vertices and [[Girth (graph theory)|girth]] 4.<ref>{{Cite OEIS |A198303 |Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth exactly g. |access-date=2023-12-15 }}</ref><ref>{{Cite OEIS |A186734 |Triangular array C(n,k) counting connected k-regular simple graphs on n vertices with girth exactly 4. |access-date=2023-12-15 }}</ref> (where the sum of all previous indexes through the seventh is 130, the total number of prime totatives of 744, including 1).<ref>{{Cite OEIS |A006924 |Number of connected trivalent graphs with 2n nodes and girth exactly 4 }}</ref>
}}

Thrice 743 is '''2229''',{{efn-lr|1=In the family of directed multigraphs that are enriched by the [[Combinatorial species|species]] of set partitions, the number of such multigraphs with four labeled directed edges (or arcs) is 2229.<ref>{{Cite OEIS |A098623 |Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs. |access-date=2024-01-20 }}</ref><ref>{{Cite journal |last=Gilbert |first=Labelle |title=Counting enriched multigraphs according to the number of their edges (or arcs) |url=https://www.sciencedirect.com/science/article/pii/S0012365X99002654?via%3Dihub |journal=Discrete Mathematics |volume=217 |issue=1–3 |publisher=[[Company:Elsevier|Elsevier]] |location=Amsterdam |year=2000 |pages=237–248 |doi=10.1016/S0012-365X(99)00265-4 |doi-access=free |mr=1766269  |zbl=0959.05061 |s2cid=22434588 }}</ref> On the other hand, {{Bullet list |2229 is equal to the sum between the sixteenth [[Lucas number]] and its counterpart non-Lucas number, plus {{math|1=1 }}: (2207, 22).<ref>{{Cite OEIS |A022801 |n-th Lucas number (A000204(n)) + n-th non-Lucas number (A090946(n+1)). |access-date=2024-01-20 }}</ref><ref>{{Cite OEIS |A000204 |Lucas numbers (beginning with 1): L(n) equal to L(n-1) + L(n-2) with L(1) equal to 1, L(2) equal to 3. |access-date=2024-01-20 }}</ref><ref>{{Cite OEIS |A090946 |Non-Lucas numbers: complement of A000204. |access-date=2024-01-20 }}</ref> |2229 is also the number of partitions of 31 with at most 3 odd parts,<ref>{{Cite OEIS |A114312 |Number of partitions of n with distinct numbers of odd and even parts. |access-date=2024-01-20 }}</ref> and the number of partitions of the sixteenth [[Composite number|composite number]] '''26'''<ref name="CompNum" /> that contain a distinct numbers of odd and even parts.<ref>{{Cite OEIS |A171967 |Number of partitions of n with distinct numbers of odd and even parts. |access-date=2024-01-20 }}</ref> |It is also the eighth member {{math|1=''n'' }} of polyhexes of size {{math|1= 6''n'' + 1 = '''49''' }} that have {{math|1=6 }}-fold [[Rotational symmetry|rotational symmetry]], preceding such polyhexes of sizes (1, 1, 2, 4, 11, 37, 136, 540) for {{math|1=0 ≤ ''n'' < 8 }};<ref>{{Cite OEIS |A253927 |Number of polyhexes (without holes) of size 6*n+1 with 6-fold rotational symmetry. |access-date=2024-01-20 }}</ref> for the number of polyhexes of size 31 ({{math|1=''n'' = 6 }}) there are thirty-seven, while for those of size 37 ({{math|1=''n'' = 5 }}) there are 136, which is the sixteenth [[Triangular number|triangular number]].<ref name="TriangNum" /> }} 2229 is also the number of [[Endomorphism#Endofunctions|endofunctions]] of [[Equivalence class|class]] <math>[5]</math> with distinct [[Cycle (graph theory)#Directed circuit and directed cycle|cycle lengths]].<ref>{{Cite OEIS |A212789 |Number of endofunctions on [n] with distinct cycle lengths. |access-date=2024-01-20 }}</ref><hr>When grouping successive composite numbers together such that the sum of the <math>n</math>–th group is a multiple of the <math>n</math>–th prime, '''2229''' is in equivalence with the twenty-fifth group-sum of composites as divided by twenty-fifth prime number, [[97 (number)|97]].<ref>{{Cite OEIS |A074127 |Group the composite numbers so that the sum of the n-th group is a multiple of the n-th prime: (4), (6), (8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22), (24, 25), (26, 27, 28, 30, 32), (33, 34, ...), ... Sequence gives the group sum divided by n-th prime for the n-th group. |access-date=2024-01-20 }}</ref> This sum occurs between '''182''' composite numbers that span the range '''('''1106, 1313''')''', without including the ''twenty-six'' prime numbers in between (starting with 1109 and ending with 1307).<br /> The first composite member of the composite grouping is '''1106''', the fourteenth 14-gonal number,<ref>{{Cite OEIS |A060354 |The n-th n-gonal number: a(n) equal to n*(n^2 - 3*n + 4)/2. |access-date=2024-01-22 }}</ref><ref>{{Cite OEIS |A051866 |14-gonal (or tetradecagonal) numbers: a(n) equal to n*(6*n-5). |access-date=2024-01-22 }}</ref> that is also the smallest {{math|1=''k'' }} such that {{math|1=<nowiki>|</nowiki>''[[Mertens function|M]]''(''k'')<nowiki>|</nowiki> = '''14''' }} (a.k.a. the inverse of the [[Mertens function]]).<ref name="InvMert">{{Cite OEIS |A051402 |Inverse Mertens function: smallest k such that <nowiki>|</nowiki>M(k)<nowiki>|</nowiki> is equal to n, where M(x) is Mertens's function A002321. |access-date=2024-01-22 }}</ref> Where 1106 is the number of regions into which the [[Plane (geometry)|plane]] is divided when drawing twenty-four [[Ellipse|ellipse]]s,<ref>{{Cite OEIS |A051890 |a(n) equal to 2*(n^2 - n + 1). |access-date=2024-01-22 }}</ref> the number of ''inequivalent'' [[Hadamard matrix]] designs of order '''24''' is 1106; explicitly, of form {{math|1='''2'''–(23, 11, 5)}}.<ref>{{Cite OEIS |A001111 |Number of inequivalent Hadamard designs of order 4n. |access-date=2024-01-22 }}</ref><ref>{{Cite web |last=Spence |first=Edward |title=Hadamard matrices |url=http://www.maths.gla.ac.uk/~es/hadamard/hadamard.php |website=Edward Spence (Homepage) |publisher=[[Organization:University of Glasgow|University of Glasgow]] Department of Mathematics |date= |access-date=2024-01-22 }}</ref> Also, the first prime number within the sequence of twenty-five consecutive composite numbers that are divisible by the twenty-fifth prime number is 1109, the '''186'''th prime number (an index that represents the thirteenth largest divisor of 744),<ref name="PNum" /> that is the seventh number such that the Mertens function reaches a new record amplitudes between zeros (of negative 15);<ref name="RecMert">{{Cite OEIS |A304241 |Indices where Mertens function A002321 reaches record amplitudes between zeros. |access-date=2024-01-22 }}</ref> the largest prime number within this sequence is the 214th prime, 1307.<ref name="PNum" /> 1106 also has a sum-of-divisors of 1920,<ref name="SigmaSoD" /> and the arithmetic mean of its divisors is 240,<ref name="ArithNum0" /><ref name="ArithNum" /> respectively the sum-of-divisors and totient of 744.<ref name="Etot" /><br />In the list of successive composite indexes — {{math|1=''A''(''n'' + 1) = ''A''(''n'') }}–th composite with {{math|1=''A''(1) = 47 }} (cf. A059408) — 1106 is part of the sequence (47, 66, 91, 122, 160, 207, '''264''', 332, '''413''', ..., '''1106''', ...); a sequence rooted in the fifteenth prime number, [[47 (number)|47]].<ref name="PNum" /> The [[Square number|square]] of 13 is 169, and the square of 14 is [[196 (number)|196]]; the latter is 1 and 3 units away from [[197 (number)|197]] and [[199 (number)|199]], which are the smallest {{math|1=''k'' }} such that {{math|1=<nowiki>|</nowiki>''M''(''k'')<nowiki>|</nowiki> = ''n'' }}, for {{math|1=''n'' }} of 7 and 8 (respectively),<ref name="InvMert" /> where the sum between the fifteenth pair of [[Twin prime|twin prime]]s<ref name="TwinPs" /> {{math|1=197 + 199 = '''396''' }} is the third [[Digit-reassembly number|digit-reassembly number]] in decimal<ref name="Osiris" /> equal to the sum of the first two (132, 264). }} whose average of divisors is 744 (as with thrice any prime number <math>3p</math>, the average of divisors will be <math>p + 1</math>).<ref name="ArithNum0" /><ref name="ArithNum" />{{efn-lr|1=Another relevant example is the ''thirtieth'' prime number 113,<ref name="PNum" /> where thrice its value is 339, whose average of divisors is '''[[114 (number)|114]]'''.<ref name="ArithNum0" /><ref name="ArithNum" /> The three largest of a total four numbers that have a sum-of-divisors of 456 (the only number to have an aliquot sum of 744)<ref name="AliSum" /> have arithmetic means that are 114 (they are 222, 302, 339, and '''407'''),<ref name="ArithNum0" /><ref name="ArithNum" /> with 339 between adjacent terms that sum to the 127th prime number and 31st [[Super-prime|super-prime]], 709.<Ref name="SupP" /> In the list of numbers to reach record amplitude values between zeroes in the Mertens function, the first few terms are (1, 31, 114, '''199''', 443, 665, 1109, ...);<ref name="RecMert" /> the fourth term is the 46th prime number (199),<ref name="PNum" /> an index that is itself the 31st in the list of composites,<ref name="CompNum" /> with 443 having an arithmetic mean of divisors of 222 (since it is prime).<ref name="ArithNum0" /><ref name="ArithNum" /> After 1109, the next term is 1637, which is 665<sub>16</sub> in [[Hexadecimal|hexadecimal]]; note that 31 is the sum-of-divisors of [[16 (number)|16]], the first of only two numbers (the other being 25).<ref name="SigmaSoD" /> Otherwise, the difference {{math|1=709 − 339 = 370 }} represents the second-largest non trivial [[Narcissistic number|Armstrong number]] equal to the sum of the cubes of their digits (of only four in [[Decimal|decimal]], aside from 0 and 1),<ref>{{Cite OEIS |A046197 |Fixed points for operation of repeatedly replacing a number with the sum of the cubes of its digits. |access-date=2024-01-23 }}</ref><ref>{{Cite OEIS |A005188 |Armstrong (or pluperfect, or Plus Perfect, or narcissistic) numbers: m-digit positive numbers equal to sum of the m-th powers of their digits. |access-date=2024-01-23 }}</ref> which follows [[153 (number)|153]], where 407 is the largest; else, 709 and 339 have a ''range'' of 371 integers, which is the third-largest Armstrong number equal to the cube of its digits.<br />The 153rd non-trivial and unitary [[Arithmetic number|arithmetic number]] after one is 222, where '''('''153, 154''')''' form the sixth Ruth-Aaron pair<ref name="RuthA">{{Cite OEIS |A006145 |Ruth-Aaron numbers (1): sum of prime divisors of n equal to the sum of prime divisors of n+1. |access-date=2024-01-23 }}</ref> with a common sum of distinct prime factors of 20<ref name="RuthAPS">{{Cite OEIS |A006146 |Sums of prime divisors of Ruth-Aaron numbers (A006145). |access-date=2024-01-23 }}</ref> (the eleventh composite, where the first Ruth-Aaron pair is between 5 and 6). Note, that the sum of the composite index of 20, alongside the twentieth composite and prime numbers, is in equivalence with {{math|1=11 + 32 + 71 = '''114''' }}. Where thrice 20 is '''60''', half 114 is '''[[57 (number)|57]]''', which represents the composite index of [[80 (number)|80]], with 57 the '''[[40 (number)|40]]'''th composite number.<ref name="CompNum" /> Ten is the sum between sums of distinct prime divisors (of 5) of the first two Ruth-Aaron pairs, where the second pair is '''('''24, 25''')''', followed by '''('''49, 50''')'''.<ref name="RuthA" /> }}

For [[Path (graph theory)#Definitions|open trail]]s of lengths eight and nine, starting and ending at fixed distinct vertices in the complete undirected graph on five labeled vertices, the number is 132 (the prime index of 743, half 264),<ref>{{Cite OEIS |A357886 |Triangle read by rows: T(n,k) is the number of open trails of length k starting and ending at fixed distinct vertices in the complete undirected graph on n labeled vertices, for n greater than or equal to 1 and k equal to 0 .. n*(n-1)/2. |access-date=2023-12-16 }}</ref> that also represents the number of [[Tree (graph theory)#Definitions|irreducible trees]] with fifteen vertices.<ref>{{Cite OEIS |A000014 |Number of series-reduced trees with n nodes. |access-date=2023-09-02 }}</ref> While for the complete undirected graph <math>K_{5}</math> there are 264 ''directed'' Eulerian circuits,<ref>{{Cite OEIS |A135388 |Number of (directed) Eulerian circuits on the complete graph K_{2n+1}. |access-date=2023-12-16 }}</ref><ref>{{Cite OEIS |A007082 |Number of Eulerian circuits on the complete graph K_{2n+1}, divided by (n-1)!^(2n+1). |access-date=2023-12-16 }}</ref> it is more specifically the number of circuits of length ten in the complete undirected graph on five labeled vertices, and as such it is the twenty-fifth element in a triangle <math>T(n,k)</math> of length <math>k</math> on <math>n</math> labeled vertices.<ref>{{Cite OEIS |A357887 |Triangle read by rows: T(n,k) is the number of circuits of length k in the complete undirected graph on n labeled vertices, for n greater than or equal to 1 and k equal to 0 .. n(n-1)/2. |access-date=2023-12-16 }}</ref>

Otherwise, 745 is the number of [[Connectivity (graph theory)|disconnected]] simple labeled graphs [[Covering graph|covering]] six vertices, where the most symmetric of these graphs has three pairs of distinct vertices each covered by one edge alone and all three edges [[Intersection graph|intersecting]]; this yields the disconnected covering graph <math>\{\{1,4\},\{2,5\},\{3,6\}\}</math> on vertices labelled <math>1</math> through <math>6</math> in a hexagonal arrangement, with the remaining 744 graphs representing all other possible conformations.<ref>{{Cite OEIS |A327070 |Number of non-connected simple labeled graphs covering n vertices. |access-date=2023-08-28 }}</ref>

456 (the only number to have an aliquot sum of 744) is the number of unlabeled ''non-mating graphs'' with seven vertices (where a mating or <math>M</math> graph is a graph where no two vertices have the same set of [[Neighbourhood (graph theory)|neighbors]]), equivalently the number of unlabeled graphs with seven vertices and at least one endpoint;<ref>{{Cite OEIS |A141580 |Number of unlabeled non-mating graphs with n vertices. |access-date=2024-01-20 }}</ref> as well as the number of [[Clique (graph theory)|cliques]] in the 7-[[Planar graph|triangular graph]], where every subset of two distinct vertices in a clique are [[Glossary of graph theory#adjacent|adjacent]].<ref>{{Cite OEIS |A290056 |Number of cliques in the n-triangular graph. |access-date=2024-01-20 }}</ref> The number of even graphs with seven vertices, where a graph is ''odd'' if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and ''even'' otherwise, is 456.<ref>{{Cite OEIS |A000568 |Number of outcomes of unlabeled n-team round-robin tournaments |access-date=2024-01-15 }}</ref><ref>{{Cite journal |last1=Royle |first1=Gordon F. |last2=Praeger |first2=Cheryl E. |last3=Glasby |first3=S.P. |last4=Freedmn |first4=Saul D. |last5=Devillers |first5=Alice |title=Tournaments and Graphs are Equinumerous |journal=Journal of Algebraic Combinatorics |volume=57 |number=2 |year=2023 |publisher=[[Company:Springer Science+Business Media|Springer]] |location=Heidelberg |pages=515–524 |doi=10.1007/s10801-022-01197-0 |doi-access=free |mr=4570411 |zbl=1512.05199 |s2cid=247958071 |arxiv=2204.01947 }}</ref>{{efn-lr|1=Within other graphs, 456 is also:{{Bullet list |The number of different [[Composition (combinatorics)|compositions]] of the [[Ladder graph|ladder graph]] <math>L_{4}</math>.<ref>{{Cite OEIS |A078469 |Number of different compositions of the ladder graph L_n. |access-date=2024-01-20 }}</ref> It is also the number of (directed) [[Hamiltonian path]]s in the {{math|1=6 }}-[[Möbius ladder]] graph,<ref>{{Cite OEIS |A137883 |Number of (directed) Hamiltonian paths in the n-Möbius ladder graph. |access-date=2024-01-20 }}</ref> and the number of directed Hamiltonian paths on the {{math|1=6 }}-[[Prism graph|prism graph]].<ref>{{Cite OEIS |A124350 |a(n) equal to 4*n*(floor(n^2/2)+1). For n greater than or equal to 3, this is the number of directed Hamiltonian paths on the n-prism graph. |access-date=2024-01-20 }}</ref> |The number of complete non-ambiguous trees of size '''5'''.<ref>{{Cite OEIS |A002190 |Sum_{n greater than or equal to 0} a(n)*x^n/n!^2 equal to -log(BesselJ(0,2*sqrt(x))). |access-date=2024-01-20 }}</ref> These trees are defined as labelled binary trees <math>T</math>, where each left (right, child) label appears once, and where a left (resp. right) label <math>U</math> ''ancestor'' to <math>V</math> is labeled strictly greater than <math>V</math>.<ref>{{Cite journal |last1=Aval |first1=Jean-Christophe |last2=Boussicault |first2=Adrien |last3=Delcroix-Oger |first3=Bérénice |last4=Hivert |first4=Florent |last5=Laborde-Zubieta |first5=Patxi |display-authors=3 |title=Non-ambiguous trees: new results and generalization |url=https://dmtcs.episciences.org/6414/pdf |journal=European Journal of Combinatorics  |volume=95 |publisher=[[Company:Elsevier|Elsevier]] |location=Amsterdam |year=2021 |pages=1–28 |doi=10.1016/j.ejc.2021.103331 |arxiv=1511.09455 |mr=4111347  |zbl=1440.05061 |s2cid=265038614 }}</ref><ref>{{Cite journal |last1=Aval |first1=Jean-Christophe |last2=Boussicault |first2=Adrien |last3=Bouvel |first3=Mathilde |last4=Silimbani |first4=Matteo |title=Combinatorics of non-ambiguous trees |journal=[[Advances in Applied Mathematics]] |volume=56 |publisher=[[Company:Elsevier|Elsevier]] |location=Amsterdam |year=2014 |pages=78–108 |doi=10.1016/j.aam.2013.11.004 |doi-access=free |mr=3194208  |zbl=1300.05127 |s2cid=14989102 |arxiv=1305.3716 }}</ref> |The number of connected regular graphs with '''18''' nodes and girth at least 5.<ref>{{Cite OEIS |A186725 |Number of connected regular graphs with n nodes and girth at least 5. |access-date=2024-01-20 }}</ref> |The number of degree {{math|1=6 }} vertices in the {{math|1='''3''' }}-[[Menger sponge]] graph.<ref>{{Cite OEIS |A367707 |Number of degree 6 vertices in the n-Menger sponge graph. |access-date=2024-01-20 }}</ref> |The number of [[Glossary of graph theory#closed|closed walks]] at a vertex of the complete graph on nine nodes <math>K_{9}</math>.<ref>{{Cite OEIS |A093134 |A Jacobsthal trisection. |access-date=2024-01-20 }}</ref> }} Otherwise, 456 appears in the <math>T</math> toothpick sequence at the twenty-second generation.<ref>{{Cite OEIS |A160172 |T-toothpick sequence (see Comments lines for definition). |access-date=2024-01-20 }}</ref> }} 

In particular, 456 is the aliquot sum of 264, the only number to have this value for <math>s(n).</math><ref name=AliSum />

=== Convolution of Fibonacci numbers ===

744 is the twelfth [[Generalizations of Fibonacci numbers#Convolved Fibonacci sequences|self-convolution]] of [[Fibonacci number]]s, which is equivalently the number of elements in all [[Subset|subset]]s of <math>\{1,2,3,...,11\}</math> with no consecutive integers.<ref>{{Cite journal |last=Moree |first=Pieter |title=Convoluted Convolved Fibonacci Numbers |url=https://cs.uwaterloo.ca/journals/JIS/VOL7/Moree/moree12.pdf |journal=[[Journal of Integer Sequences]] |volume=7 |issue=2 |publisher=University of Waterloo David R. Cheriton School of Computer Science |location=Waterloo, Ont., CA |year=2004 |page=13 (Article 04.2.2) |bibcode= 2004JIntS...7...22M |arxiv=math.CO/0311205 |mr=2084694 |s2cid=14126332 |zbl=1069.11004 }}</ref><ref name="ConvFib">{{Cite OEIS |A001629 |Self-convolution of Fibonacci numbers. |access-date=2023-07-16 }}</ref><ref>{{Cite journal |last1=Belbachir |first1=Hacène |last2=Djellal |first2=Toufik |last3=Luque |first3=Jean-Gabriel |title=On the self-convolution of generalized Fibonacci numbers |url=https://www.tandfonline.com/doi/abs/10.2989/16073606.2022.2043949 |journal=Quaestiones Mathematicae |publisher=Taylor & Francis |location= Oxfordshire, UK |volume=46 |issue=5 |year=2023 |pages=841–854 |doi=10.2989/16073606.2022.2043949 |mr=4592901 |arxiv=1703.00323 |s2cid=119150217 |zbl=07707543 }}</ref>

== Abstract algebra ==

The [[J-invariant|'''''j'''''–invariant]] holds as a [[Fourier series]] [[Modular form#Modular functions|{{math|1=''q'' }}–expansion]],{{efn-lg|1=The {{math|1='''''j''''' }}–invariant can be computed using [[Eisenstein series]] {{math|1=''E''<sub>4</sub> }} and {{math|1=''E''<sub>6</sub> }}, such that:<br /> {{math|1='''''j'''''(𝜏) = 1728 {{sfrac|''E''<sub>4</sub>(𝜏)<sup>3</sup>|''E''<sub>4</sub>(𝜏)<sup>3</sup> − ''E''<sub>6</sub>(𝜏)<sup>2</sup> }} , }}<br /> where {{math|1=''E''<sub>4</sub>(𝜏) = 1 + 240 '''Σ'''{{su|b=<sub>''n''{{hair space}}={{hair space}}1</sub>|p=∞}}{{resize|150%|(}}{{sfrac|''n''<sup>3</sup>''q''<sup>''n''</sup>|1 − ''q''<sup>''n''</sup> }}{{resize|150%|)}} }} and {{math|1=''E''<sub>6</sub>(𝜏) = 1 − 504 '''Σ'''{{su|b=<sub>''n''{{hair space}}={{hair space}}1</sub>|p=∞}}{{resize|150%|(}}{{sfrac|''n''<sup>5</sup>''q''<sup>''n''</sup>|1 − ''q''<sup>''n''</sup> }}{{resize|150%|)}} }}, with {{math|1=''q'' = exp(2π''i''𝜏) }}.<br />The respective {{math|1=''q'' }}–expansions of these two Eisenstein series have coefficients whose numerical values are in proportion to 240 and −504, respectively;<ref>{{Cite OEIS |A004009 |Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice. |access-date=2023-10-05 }}</ref><ref>{{Cite OEIS |A013973 |Expansion of Eisenstein series E_6(q) (alternate convention E_3(q)). |access-date=2023-10-05 }}</ref> where specifically the sum and difference between the absolute values of these two numbers is {{math|1=240 + 504 = 744 }} and {{math|1=504 − 240 = 264 }}.<br />Furthermore, when considering the only smaller even (here, ''non-modular'') series {{math|1=''E''<sub>2</sub>(𝜏) = 1 − 24 '''Σ'''{{su|b=<sub>''n''{{hair space}}={{hair space}}1</sub>|p=∞}}{{resize|150%|(}}{{sfrac|''nq<sup>n</sup>''|1 − ''q<sup>n</sup>'' }}{{resize|150%|)}} }}, the sum between the absolute value of its constant multiplicative term (24) and that of {{math|1=''E''<sub>4</sub>(𝜏) }} (240) is equal to '''264''' as well. The 16th coefficient in the expansion of {{math|1=''E''<sub>2</sub>(𝜏) }} is −744, as is its 25th coefficient.<ref>{{Cite OEIS |A006352 |Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2). |access-date=2023-10-05 }}</ref><br />Alternatively, the '''''j'''''–invariant can be computed using a sextic polynomial as: {{math|1=''j''(''λ'') = 2<sup>8</sup> × {{sfrac|(''λ''<sup>2</sup> − ''λ'' + 1)<sup>3</sup> |''λ''<sup>2</sup>(''λ'' − 2)<sup>2</sup> }} }} where {{math|1='''λ''' }} represents the ''λ''−modular function, with {{math|1=256 = 2<sup>8</sup> }}.<ref>{{Cite book |last=Hartshorne |first=Robin |title=Algebraic Geometry |chapter=Chapter 4: Curves |url=https://link.springer.com/book/10.1007/978-1-4757-3849-0 |series=[[Graduate Texts in Mathematics]] |volume=52 |publisher=[[Company:Springer Science+Business Media|Springer-Verlag]] |location=Berlin |year=1977 |pages=316–321 |isbn=978-1-4419-2807-8 |doi=10.1007/978-1-4757-3849-0 |oclc=13348052 |mr=1567149 |zbl=0367.14001 |s2cid=5695770 }}</ref> }} <math display=block>j(\tau) = q^{-1} + 744 + 196\,884 q + 21\,493\,760 q^2 + 864\,299\,970 q^3 + \cdots</math>

The [[Monster group|Friendly Giant]] <math>\mathrm {F_{1}}</math> contains an infinitely [[Graded vector space|graded]] faithful [[Faithful representation|dimensional representation]] equivalent to the <math>q</math> coefficients of this series, where <math>q = e^{2\pi i\tau}</math> and <math>\tau</math> the [[Half-period ratio|half-period ratio]] of an [[Elliptic function|elliptic function]].<ref>{{Cite journal |doi=10.4153/CMB-1999-050-1 |doi-access=free |first1=Bruce C. |last1=Berndt |first2=Heng Huat |last2=Chan |title=Ramanujan and the modular j-invariant |journal=[[Canadian Mathematical Bulletin]] |volume=42 |issue=4 |year=1999 |pages=427–440 |mr=1727340 |s2cid= 1816362 |zbl=}}</ref>

Also, the [[Almost integer|almost integer]]<ref>{{Cite OEIS |A060295 |Decimal expansion of exp(Pi*sqrt(163)). |access-date=2023-08-18 }}</ref>

<math display=block>e^{\pi \sqrt{163}} \approx 262\,537\,412\,640\,768\,743.999\,999\,999\,999\,250\,072\,59 \approx 640\,320^3 + 744.</math>

This number is known as [[Ramanujan's constant]], which is [[Transcendental number|transcendental]].<ref>{{Cite book |last=Barrow |first=John D. |title=The Fundamental Constants |chapter=The Constants of Nature |chapter-url=https://archive.org/details/constantsofnatur0000barr_t1y7/ |publisher=Jonathan Cape |page=72 |year=2002 |location=London |doi=10.1142/9789812818201_0001 |chapter-url-access=registration |isbn = 0-224-06135-6 |s2cid=125272999 }}</ref> [[Biography:Mark Ronan|Mark Ronan]] and other prominent mathematicians have noted that the appearance of <math>163</math> in this number is relevant within '''moonshine theory''', where one hundred and sixty-three is the largest of nine [[Heegner number]]s that are [[Square-free integer|square-free positive integer]]s <math>d</math> such that the imaginary [[Quadratic field|quadratic field]] <math display="inline">\Q\left[\sqrt{-d}\right]</math> has [[Ideal class group|class number]] of <math>1</math> (equivalently, the [[Ring of integers|ring of integers]] over the same [[Algebraic number field|algebraic number field]] have unique factorization).<ref name=MarkR>{{Cite book |last1=Ronan |first1=Mark |author-link=Mark Ronan |title=Symmetry and the Monster: One of the Greatest Quests of Mathematics |publisher=[[Organization:Oxford University Press|Oxford University Press]] |url=https://archive.org/details/symmetrymonstero0000rona/ |url-access=registration |location=New York |year=2006 |pages=1–255 |isbn=978-0-19-280722-9 |oclc=180766312 |mr=2215662 |zbl=1113.00002 }}</ref>{{rp|pp.227,228}} <math>\mathrm {F_{1}}</math> has one hundred and ninety-four ([[194 (number)|194]]) [[Conjugacy class|conjugacy class]]es generated from its [[Character table|character table]] that collectively produces the same number of elliptic ''moonshine functions'' which are not all linearly independent; only one hundred and sixty-three are entirely independent of one another.<ref>{{Cite web |last=Ronan |first=Mark |author-link=Mark Ronan |title=163 and the Monster |url=http://www.markronan.com/mathematics/symmetry-corner/163-and-the-monster/ |website=Mark Ronan |date=6 March 2013 |access-date=2023-08-17 }}</ref> The linear [[Errors and residuals|term of error]] <math>O</math> for Ramanujan's constant is approximately,

<math display=block>\frac{-196\,884}{e^{\pi \sqrt{163}}} \approx \frac{-196\,884}{640\,320^3+744} \approx -0.000\,000\,000\,000\,75 ,</math>

where <math>196\,884</math> is the value of the minimal [[Faithful representation|faithful complex dimensional representation]] of <math>\mathrm {F_{1}}</math>, the largest [[Sporadic group|sporadic group]].<ref name="JanChr">{{Cite journal |last=Jansen |first=Christoph |date=2005 |title=The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups |journal=[[LMS Journal of Computation and Mathematics]] |volume=8 |pages=122−123 |publisher=London Mathematical Society |location=London |doi=10.1112/S1461157000000930 |doi-access=free |s2cid=121362819 |zbl=1089.20006 |mr=2153793 }}</ref>

Specifically, all three common prime factors <math>(2,3,5)</math> that divide the Euler totient, sum-of-divisors, and reduced totient of <math>744</math> are the smallest and only primes that divide the orders of all twenty-six sporadic groups, in contrast with only six groups <math>(\mathrm {F_{1}},\text{ }\mathrm {B},\text{ }\mathrm {F_{3}},\text{ }\mathrm {Ly},\text{ }\mathrm {ON},\text{ }\mathrm {J_{4}})</math> whose orders can be divided by the largest consecutive supersingular prime and largest prime factor of seven hundred and forty-four, <math>31</math>;<ref name=MarkR />{{rp|pp.244–246}}{{efn-lg|1=The smallest sporadic group is [[Mathieu group]] {{math|1=[[Mathieu group M11|'''M'''<sub>11</sub>]] }}, with an irreducible complex representation in ten dimensions,<ref name="JanChr" /> that is one of five ''[[Mathieu group|first generation]]'' sporadic groups.<ref name="Gr1998">{{Cite book |last=Griess, Jr. |first=Robert L. |title=Twelve Sporadic Groups |series=Springer Monographs in Mathematics |publisher=[[Physics:Springer-Verlag|Springer-Verlag]] |location=Berlin |year=1998 |pages=54–79 |isbn=9783540627784 |oclc=38910263 |mr=1707296 |zbl=0908.20007 }}</ref> Its group order is {{math|1=7920 = 8 × 9 × 10 × 11 }} (one less than the 10<sup>3</sup> indexed prime number),<ref name=MarkR />{{rp|pp.244–246}}<ref name="PNum" /> with a Euler totient of {{math|1=1920 = σ(744) }}.<ref name="Etot" /> {{math|1='''M'''<sub>11</sub> }} also holds a lowest [[Five-dimensional space|five-dimensional]] faithful representation over the field with three elements, the lowest of any such group.<ref name="JanChr" /> On the other hand, {{math|1=[[Mathieu group M12|'''M'''<sub>12</sub>]] }} is the second-largest Mathieu group and sporadic group by order,<ref name=MarkR />{{rp|pp.244–246}} and more specifically the thirty-first largest non-[[Cyclic group|cyclic group]] (where {{math|1='''M'''<sub>11</sub> }} is the fifteenth largest),<ref name="OrdNCG">{{Cite web |last=Madore |first=David Alexander |title=Orders of non abelian simple groups |url=http://www.madore.org/~david/math/simplegroups.html |website=www.madore.org |date=22 January 2003 |access-date=2024-01-10 }}</ref><ref name="OEISONCG">{{Cite OEIS |A109379 |Orders of non-cyclic simple groups (with repetition). |access-date=2024-01-10 }}</ref> with group order {{math|1=2<sup>6</sup> × 3<sup>3</sup> × 5 × 11 = 95040 }} whose Euler totient (23040)<ref name="Etot" /> is divisible by 1920 twelve-times over (as its 50th largest divisor greater than 1). The nineteenth and twentieth largest non-cyclic groups are {{math|1='''''A'''''<sub>3</sub>(2) ≃ '''A'''<sub>8</sub> }} and {{math|1='''''A'''''<sub>2</sub>(4) }} of orders {{math|1=20160 = 2<sup>6</sup> × 3<sup>2</sup> × 5 × 7 }}, the latter with larger [[Outer automorphism group|outer automorphism group]] {{math|1='''D'''<sub>12</sub> }}; '''20160''' is the twenty-third [[Highly composite number|highly composite number]],<ref>{{Cite OEIS |A002182 |Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record. |access-date=2024-01-09 }}</ref> that is divisible by both 960 (the Zumkeller half of 744),<ref name="Zumk" /> and '''5040''' (the largest of twenty-seven numbers in Robin's theorem to hold [[Inequality (mathematics)|inequality]] {{math|1=σ(''n'') ≥ ''e''<sup>γ</sup>''n''loglog''n'' }}, see latter points discussing this sequence).<ref name="RobTh" /> It is one less than the largest number (20161, the '''1456'''th indexed number) which cannot be represented as the sum of two [[Abundant number|abundant number]]s.<ref name="NSumAbN">{{Cite OEIS |A048242 |Numbers that are not the sum of two abundant numbers (not necessarily distinct). |access-date=2024-01-09 }}</ref> Otherwise, the [[Tits group]] {{math|1='''T''' }}, the only finite simple group that can classify as of [[Group of Lie type|Lie type]] or sporadic, fits as the fifth-largest sporadic group, whose group order {{math|1=2<sup>11</sup> × 3<sup>3</sup> × 5<sup>2</sup> × 13 = 17971200 }} holds a totient of 4423680, divisible by the sum-of-divisors of seven-hundred and forty-four 2304 times (where its 62nd largest divisor is 1728, followed by 1920).<br />Of all (here, twenty-seven) sporadic groups, only the seventh largest [[Janko group]] {{math|1='''[[Janko group J3|J<sub>3</sub>]]''' }} has an order, {{math|1=<nowiki>|</nowiki>'''J<sub>3</sub>'''<nowiki>|</nowiki> = 2<sup>7</sup> × 3<sup>5</sup> × 5 × 17 × 19 = 50232960 }},<ref name=MarkR />{{rp|pp.244–246}} whose totient value is not divible by 1920: {{math|1=11943936 ÷ 1920 = '''6022.8''' }}; where specifically its group order has a sum of prime factors (inclusive of multiplicities) equal to '''70'''.<br />Where {{math|1='''M'''<sub>11</sub> }} and {{math|1='''M'''<sub>12</sub> }} are respectively the fifteenth and thirty-first largest indexed non-cyclic groups by group order, these indices sum to {{math|1=15 + 31 = 46}}, where the third-largest Mathieu group and fourth-largest sporadic group {{math|1='''[[Mathieu group M22|M]]'''<sub>22</sub> }} is the forty-sixth such largest non-cyclic group.<ref name="OrdNCG" /><ref name="OEISONCG" /> [[46 (number)|46]] is the largest ''[[Parity (mathematics)|even]]'' number that cannot be represented as the sum of two abundant numbers,<ref name="NSumAbN" /> that is also equal to the total number of [[Maximal subgroup|maximal subgroup]]s of the [[Monster group#Maximal subgroups|Friendly Giant]] {{math|1=<math>\mathbb{F}</math><sub>1</sub> }},<ref>{{cite arXiv |title=The maximal subgroups of the Monster |last1=Dietrich |first1=Heiko|last2=Lee |first2=Melissa |last3=Popiel |first3=Tomasz |class=math. GR |pages=1–23 |date=6 December 2023 |eprint=2304.14646 }} {{Bibcode|2023arXiv230414646D}}.</ref> which collectively holds 20 of 26 sporadic groups as [[Subquotient|subquotient]]s (strictly).<ref name=MarkR />{{rp|pp.244–246}}<br />'''46''' is the thirty-first composite,<ref name="CompNum" /> whose ''aliquot sum'' is '''26''', the only number to hold this value.<ref name="AliSum" />}} three of these belong inside the small family of six [[Pariah group|pariah group]]s that are not [[Subquotient|subquotient]]s of <math>\mathrm {F_{1}}.</math><ref>{{Cite journal |last=Griess, Jr. |first=Robert L. |title=The Friendly Giant |url=https://www.digizeitschriften.de/dms/img/?PPN=PPN356556735_0069&DMDID=dmdlog7 |journal=Inventiones Mathematicae |volume=69 |year=1982 |pages=91−96 |doi=10.1007/BF01389186 |bibcode=1982InMat..69....1G |hdl=2027.42/46608 |hdl-access=free |mr=671653 |s2cid=123597150 |zbl=0498.20013 }}</ref> The largest [[Supersingular prime (moonshine theory)|supersingular prime]] that divides the [[Order (group theory)|order]] of <math>\mathrm {F_{1}}</math> is <math>71,</math><ref>{{Cite book |last=Ogg |first=A. P. |editor1-last=Cooperstein |editor1-first=Bruce |editor2-last=Mason |editor2-first=Geoffrey |chapter=Modular Functions |title=The Santa Cruz Conference on Finite Groups. |series=Proceedings of Symposia in Pure Mathematics |location=Providence, RI |publisher=American Mathematical Society |volume=37 |pages=521–532 |year=1981 |isbn=0-8218-1440-0 |doi=10.1090/PSPUM/037 |mr=0604631 |zbl=0443.00007 }}</ref><ref>{{Cite OEIS |A002267 |The 15 supersingular primes: primes dividing order of Monster simple group. |access-date=2023-09-26 }}</ref> which is the eighth self-convolution of Fibonacci numbers, where <math>744</math> is the twelfth.<ref name="ConvFib" />{{efn-lg|1= The sequence of self-convoluted Fibonacci numbers starts {0, 0, 1, 2, 5, 10, 20, 38, '''71''', 130, 235, 420, '''744''', 1308, 2285, 3970, 6865, 11822...}.<ref name="ConvFib" /> The sum of the first seven terms (from the zeroth through the sixth term) is equal to 38, which is equivalent to the {{math|1=a(n) = 7}} member in this sequence. Taking the sum of the three terms that lie between 71{{efn-lr|1='''[[71 (number)|71]]''' is the twentieth prime number and 31 the eleventh. In turn, [[20 (number)|20]] is the eleventh ''composite number''<ref name="CompNum" /> that is also the sixth self-convolution of Fibonacci numbers before 38, which is the prime index of 163.<br />71 is also part of the largest pair of [[Brocard's problem#Brown numbers|Brown numbers]] {{math|1=(71, 7) }}, of only three such pairs; where in its case {{math|1=7<sup>2</sup> − 1 = 5040 }}.<ref>{{Cite OEIS |A216071 |Brocard's problem: positive integers m such that m^2 equal to n! + 1 for some n. |access-date=2023-10-10 }}</ref><ref>{{Cite OEIS |A085692 |Brocard's problem: squares which can be written as n!+1 for some n. |access-date=2023-10-10 }}</ref> Consequently, both 5040 and 5041 can be represented as sums of non-consecutive [[Factorial|factorial]]s, following 746, 745, and 744;<ref name="N-CF" /> where {{math|1=5040 + 5041 = 10081}} holds an aliquot sum of 611, which is the ''composite index'' of 744.<ref name="CompNum" /><br />'''[[5040 (number)|5040]]''' is the nineteenth [[Superabundant number|superabundant number]]<ref>{{Cite OEIS |A004394 |Superabundant [or super-abundant] numbers: n such that sigma(n)/n greater than sigma(m)/m for all m less than n, sigma(n) being A000203(n), the sum of the divisors of n. |access-date=2023-10-10 }}</ref> that is also the largest factorial that is a [[Highly composite number|highly composite number]],<ref>{{Cite OEIS |A002182 |Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record. |access-date=2023-10-10 }}</ref> and the largest of twenty-seven numbers {{math|1=''n'' }} for which the [[Inequality (mathematics)|inequality]] {{math|1=σ(''n'') ≥ ''e''<sup>γ</sup>''n''loglog''n'' }} holds, where {{math|1='''γ''' }} is the [[Euler–Mascheroni constant]]; this inequality is shown to fail for all larger numbers if and only if the [[Riemann hypothesis]] is ''true'' (known as '''[[Divisor function#Growth rate|Robin's theorem]]''').<ref name="RobTh">{{Cite OEIS |A067698 |Positive integers such that sigma(n) greater than or equal to exp(gamma) * n * log(log(n)). |access-date=2023-10-09 }}</ref> 5040 generates a sum-of-divisors {{math|1='''19344''' = 13 × 31 × 48 }} that itself contains four divisors in proportion with 744 (and therefore, divisors also in proportion with 248 as well); which makes it one of only two numbers out of these twenty-seven integers {{math|1=''n'' }} in Robin's theorem to hold {{math|1=σ(''n'') }} such that {{math|1=744''m'' }} {{pipe}} {{math|1= σ(''n'') }} for any subset of divisors {{math|1=''m'' }} of {{math|1=''n'' }}; the only other such number is 240:<ref name="RobThRieH" /> {{Bullet list |{{math|1=19344 ÷ 26 = 744 }} |{{math|1=9672 ÷ 13 = 744 }} |{{math|1=1488 ÷ 2 = 744 }} }} Where also {{math|1=19344 ÷ 78 = 248 }}, with 248 and 744 respectively as the 24th and 30th largest divisors (where in between these is {{math|1=403 = 13 × 31 }}, which is the middle indexed composite number congruent {{math|1=±1 mod 6 }} less than 744 that is not part of its composite totatives); 2418, the 35th largest, is the seventh number {{math|1=''n'' }} after 744 such that {{math|1=''σ''(''φ''(''n'')) }} is {{math|1=''n'' }}.<ref name=SigmPhi /> Furthermore, in this sequence of integers in Robin's theorem, between 240 and 5040 lie four numbers, where the sum between the first three of these {{math|1=360 + 720 + 840 = 1920 }} is in equivalence with {{math|1=σ(744). }} The first number to be divisible by all positive non-zero integers less than 11 is the penultimate number in this sequence [[2520 (number)|2520]], where {{math|1=2520 − 840 − 720 = 960 }} represents a ''Zumkeller half'' from the set of divisors of 744,<ref name="Zumk" /> with {{math|1=σ(720) = 2418 }}<ref name="AliSum" /> (and while {{math|1=720 + 24 = 744 = 6! + 24 }}, where 720 is the smallest number with thirty divisors,<ref name="LeastD" /> equal to {{math|1=1176 − 456 }}, a difference between the aliquot sum of 744, and the only number to have an aliquot sum of 744; wherein 720 is also in equivalence with {{math|1=σ(264) = 456 + 264 }}).<ref name="AliSum" /><ref name="SigmaSoD" /> {{math|1=5040 = 7! = 10 × 9 × 8 × 7 }} is divisible by the first twelve non-zero integers, except for 11. }} and 744 (i.e. 130, 235, 420) yields 785, whose [[Aliquot sum|aliquot sum]] is '''163''',<ref name="AliSum" />{{efn-lr|1='''163''' is also the sum between prime indexes '''('''[[81 (number)|81]], [[82 (number)|82]]''')''' that belong to numbers adjacent to the twenty-second [[Interprime|interprime]] {{math|1=[[420 (number)|420]] = 101 + 103 + 107 + 109 }}, the twenty-sixth number that is the sum of four consecutive primes<ref>{{Cite OEIS |A034963 |Sums of four consecutive primes. |access-date=2024-01-16 }}</ref> that is also the 45th number to return {{math|1=0 }} for the Mertens function.<ref name="Mert0" /> It is the sixth such interprime to yield prime values from sums of prime indices of adjacent numbers, after (138, 72, 12, 6, 4) and preceding 432, the abundancy of 744.<ref name="AbunoN" /> 420, with a Euler totient of 96,<ref name="Etot" /> is half of [[840 (number)|840]], the smallest number with exactly thirty-two divisors,<ref name="LeastD" /> and twice [[210 (number)|210]], the fourth [[Primorial|primorial]] {{math|1=(2 × 3 × 5 × 7) }}, where the previous three primorials sum to {{math|1=2 + 6 + 30 = [[38 (number)|38]] }},<ref name="Primor">{{Cite OEIS |A002110 |Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#. |access-date=2024-01-16 }}</ref> in equivalence with the ''prime index'' of 163.<ref name="PNum" /> Furthermore, the '''163'''rd ''composite number'' is 210,<ref name="CompNum" /> where 210 is thrice [[70 (number)|70]]. Seventy is the fifth pentatope number,<ref>{{Cite OEIS |A000332 |Binomial coefficient binomial(n,4) equal to n*(n-1)*(n-2)*(n-3)/24. |access-date=2024-01-16 }}</ref> that represents the sum of prime factors (inclusive of multiplicities) in what is described as the ''least-interrelated'' sporadic group, Janko group {{math|1='''J'''<sub>3</sub> }}, vis-à-vis all other sporadic groups (including the pariah groups), and their relevant algebraic structures. 70 is primarily used in the construction of the [[Leech lattice]] in twenty-four dimensions, that involves the only non-trivial solution (i.e., '''1''') to the [[Cannonball problem|cannonball problem]], equal to the sum of the squares of the first twenty-four integers; 70 is also equal to the number of [[Conformal field theory|conformal field theories]] in the form of [[Vertex operator algebra|vertex operator algebra]]s with [[Physics:Central charge|central charge]] of {{math|1=24 }} with weight {{math|1=1 }} (and aside from {{math|1=''V''<sub>0</sub> }}, see {{slink||E8 and the Leech lattice}}), that are rooted in the Leech lattice, of which the largest two exist in dimension seven hundred and forty-four. 210 is, furthermore, the first non-trivial '''71'''-gonal number of the form {{math|1=''P''(''s'',''n'') = {{sfrac|(''s'' − 2)''n''<sup>2</sup> − (''s'' − 4)''n'' |2 }} }} (for the {{math|1=''n'' }}th polygonal number with {{math|1=''s'' }} sides), that precedes 418, the sum of integers between 13 and 31, inclusive. Importantly, 210 is the largest number {{math|1=''n'' }} where the number of distinct representations of {{math|1=''n'' }} as the [[Goldbach's conjecture|sum of two primes]] is at most the number of primes in the interval {{math|1={{resize|150%|(}}{{sfrac|''n''|2}}, ''n'' – 2{{resize|150%|)}} }}.<ref>{{cite journal |last1=Deshouillers |first1=Jean-Marc |author-link1=Jean-Marc Deshouillers |last2=Granville |first2=Andrew |author-link2=Andrew Granville |last3=Narkiewicz |first3=Władysław |last4=Pomerance |first4=Carl |author-link4=Carl Pomerance |title=An upper bound in Goldbach's problem |journal=[[Mathematics of Computation]] |publisher=American Mathematical Society |location=Washington, D.C. |date=1993 |volume=61|issue=203 |pages=209–213 |doi=10.1090/S0025-5718-1993-1202609-9 |bibcode=1993MaCom..61..209D |doi-access=free |mr=1202609 |zbl=0780.11047 |s2cid=51831845 }}</ref><hr>'''420''', which is the eleventh self-convolution of Fibonacci numbers,<ref name="ConvFib" /> preceding 744, is the totient value of '''473''', the first non-trivial number to hold this value (after 421). Twice 744 is '''1488''', which lies between the '''50'''th pair of twin primes '''('''1487, 1489'''),'''<ref name="TwinPs" /> whose successive prime indices (236, 237) ''add'' to 473. 1488 is the '''235'''th average of successive odd primes,<ref name="AvgCP" /> where 235 is the sixth self-convolution of Fibonacci numbers, preceding 420.<ref name="ConvFib" /> The fiftieth composite number, on the other hand, is '''70'''.<ref name="CompNum" />}} the thirty-eighth prime number.<ref name="PNum" /> 785 is the 60th number to return {{math|1=0 }} for the [[Mertens function]], which also includes 163, the 13th such number.<ref name="Mert0" /> 785 is also the number of [[Tree (graph theory)|irreducible]] [[Tree (graph theory)#Rooted tree|planted trees]] (of root vertex having degree one) with six [[Tree (graph theory)#Tree|leaves]] of two [[Graph coloring|colors]].<ref>{{Cite OEIS |A050381 |Number of series-reduced planted trees with n leaves of 2 colors. |access-date=2023-09-02 }}</ref> }}

The largest three Heegner numbers with <math>d > 19</math> also give rise to ''almost integers'' of the form <math>e^{\pi \sqrt{d}}</math> which involve <math>744</math>. In increasing [[Order of approximation|orders of approximation]],<ref name="KlaiJan">{{Cite thesis |last=Klaise |first=Janis |title=Orders in Quadratic Imaginary Fields of small Class Number. |url=https://warwick.ac.uk/fac/cross_fac/complexity/people/students/dtc/students2013/klaise/janis_klaise_ug_report.pdf |degree=MMath |publisher=[[Organization:University of Warwick|University of Warwick]] Centre for Complexity Science |year=2012 |pages=1–24 |s2cid=126035072 }}</ref>{{rp|p.20–23}}{{efn-lg|1=163 is the thirty-eighth indexed prime and [[67 (number)|67]] the nineteenth (with the biprime {{math|1=38 = 2 × [[19 (number)|19]]) }}, for values {{math|1=''d'' }}. In the approximation for the almost integer containing the fourteenth prime number as the seventh Heegner number [[43 (number)|43]] for {{math|1=''d''}}, 960 is the smallest number with exactly twenty-eight divisors,<ref name="LeastD" /> equivalent to the sum of either of two sets of divisors of 744 that collectively add to {{math|1=1920 }}, as mentioned in {{slink||Totients}}. Likewise, in the approximation for the almost integer containing the eighth Heegner number 67 for {{math|1=''d''}}, 5280 is equal to the sum between '''240''' and '''5040''', which are the only two numbers in the set of twenty-seven integers in Robin's theorem for the Riemann hypothesis that have a set of divisors whose sums are divisible by 744;<ref name="RobThRieH">{{Cite journal |last1=Caveney |first1=Geoffrey |last2=Nicolas |first2=Jean-Louis |last3=Sondow |first3=Jonathan |title=Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis |url=http://math.colgate.edu/~integers/l33/ |journal=Integers |volume=11 |publisher=[[Company:De Gruyter|De Gruyter]] |location=Berlin  |year=2011 |issue=6 |pages=755 (A33) |doi=10.1515/INTEG.2011.057 |mr=2798609 |zbl=1235.11082 |s2cid=14573608 |arxiv=1110.5078 }}</ref> 5280 also lies between the 131st pair of twin primes {{math|1=(5281, 5279) }},<ref name="TwinPs" /> respectively the 701st and 700th prime numbers, where 5281 is the 126th ''[[Super-prime|super-prime]]''.<ref name="SupP" />{{efn-lr|1=Worth noting, 126 is the sum of prime indices of the first class of three-digit [[Permutable prime|permutable prime]]s in decimal {{math|1='''([[113 (number)|113]]''', '''[[131 (number)|131]]''', '''[[311 (number)|311]])''' }},<ref>{{Cite OEIS |A258706 |Absolute primes: every permutation of digits is a prime. Only the smallest representative of each permutation class is shown. |access-date=2023-12-13 }}</ref> respectively the thirtieth, thirty-second, and sixty-fourth prime numbers<ref name="PNum" /> (where [[64 (number)|64]] is twice [[32 (number)|32]], and the 45th composite, with [[45 (number)|45]] itself the 30th composite);<ref name="CompNum" /> and this sum of prime numbers is equivalent with '''[[555 (number)|555]]'''. Furthermore, 126 is the seventh successive composite index starting with 13 (ie., 13, '''22''', 34, 50, 70, 95, '''126''').<ref>{{Cite OEIS |A059408 |a(n+1) as the a(n)-th composite and a(1) equal to 13. |access-date=2024-01-19 }}</ref><br />''Note, too,'' that '''[[126 (number)|126]]''' is equal to the sum of the composite index of [[22 (number)|22]] ([[13 (number)|13]]), and the twenty-second composite and prime numbers {{math|1=([[34 (number)|34]], [[79 (number)|79]]) }}, whose sum generated between the latter two is '''113'''. 13 and 79 are also first members in their respective paired permutable primes in base ten {{math|1='''(13''', '''31)''' }} and {{math|1='''(79''', '''97)''' }}, respectively the smallest and largest two-digit pairs, where '''11''' is the only two-digit permutable prime without a distinct dual pair with differing permuting digits.<ref name="PermP" /><br />Where '''555''' is generated from the first class of three-digit permutable primes in decimal, and ''prime and composite indices associated'' with '''22''', 131 and 126 sum to '''[[257 (number)|257]]''', which is the '''[[55 (number)|55]]'''th indexed prime<ref name="PNum" /> and tenth prime number that is not a cluster prime<ref name="ClustP" /> (and where, 131<sub>10</sub> has a [[Digit sum|digit sum]] of '''5''').<br />Also, {{math|1=126 + 125 = 351 }}, a value in equivalence with the twenty-sixth triangular number,<ref name="TriangNum" /> with 125 the cube of '''[[5|5]]'''.}} }}

<math display=block>\begin{align}
e^{\pi \sqrt{19}}  &\approx {\color{white}000\,0}96^3+744-0.22\\
e^{\pi \sqrt{43}}  &\approx {\color{white}000\,}960^3+744-0.000\,22\\
e^{\pi \sqrt{67}}  &\approx {\color{white}00}5\,280^3+744-0.000\,0013\\
e^{\pi \sqrt{163}} &\approx 640\,320^3+744-0.000\,000\,000\,000\,75
\end{align}
</math>

Square-free positive integers over the negated imaginary quadratic field with class number of <math>2</math> also produce almost integers for values of <math>d</math>, where for instance there is <math>199\,148\,648^2 - 0.000\,97\ldots \approx e^{\pi \sqrt{148}} + (8 \times 10^{3} + 744).</math><ref>{{Cite web |last=Weisstein |first=Eric W. |title=Almost integer |website=[[MathWorld]] -- A [[Physics:WolframAlpha|WolframAlpha]] Resource |url=https://mathworld.wolfram.com/AlmostInteger.html |access-date=2023-10-18 }}
:The list below gives numbers of the form {{math|1=''x'' ≡ ''e''<sup>''π'''''√'''''n''</sup> }} for {{math|1=''n'' ≤ 1000 }} for which {{math|1={{pipe}}[[Floor and ceiling functions#Rounding|nint]](''x'') − ''x''{{pipe}} ≤ 10<sup>−3</sup> }}.
{| class=wikitable; align="center"
! style="text-align:center | {{math|1=''n'' }} 
! style="text-align:center | {{math|1={{pipe}}nint(''x'') − ''x''{{pipe}} }}
|-
| style="text-align: center" |'''25'''
| style="text-align: center" | −0.00066
|-
| style="text-align: center" |37
| style="text-align: center" |−0.000022
|-
| style="text-align: center" |43
| style="text-align: center" |−0.00022
|-
| style="text-align: center" |58
| style="text-align: center" |−1.8×10<sup>−7</sup>
|-
| style="text-align: center" |67
| style="text-align: center" |−1.3×10<sup>−6</sup>
|-
| style="text-align: center" |74
| style="text-align: center" |−0.00083
|-
| style="text-align: center" |'''148'''
| style="text-align: center" |0.00097
|-
| style="text-align: center" |163
| style="text-align: center" |−7.5×10<sup>−13</sup>
|-
| style="text-align: center" |232
| style="text-align: center" |−7.8×10<sup>−6</sup>
|-
| style="text-align: center" |268
| style="text-align: center" |0.00029
|-
| style="text-align: center" |522
| style="text-align: center" |−0.00015
|-
| style="text-align: center" |652
| style="text-align: center" |1.6×10<sup>−10</sup>
|-
| style="text-align: center" |'''719'''
| style="text-align: center" |−0.000013
|-

=== D<sub>4</sub> and F<sub>4</sub> ===

<math>744</math> is theta series coefficient <math>25</math> of four-dimensional [[16-cell honeycomb|cubic lattice]] <math>\mathbb {D}_{4} \cong  \mathbb {(Z^{4})^{+}}.</math><ref name="ThD4">{{Cite OEIS |A004011 |Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}. |access-date=2023-12-23 }}</ref><ref name="JiH">{{Cite book |last=Chun |first=Ji Hoon |chapter=Sphere Packings [2013] |chapter-url=https://math.osu.edu/sites/math.osu.edu/files/SpherePackings.pdf |editor1-last=Tsfasman |editor1-first=Michael |editor2-last=Vlǎduţ |editor2-first=Serge |editor3-last=Nogin |editor3-first=Dmitry |display-editors=1 |title=Algebraic Geometry Codes: Advanced Chapters |series=Mathematical Surveys and Monographs |publisher=American Mathematical Society |location=Providence, RI |volume=238 |year=2019 |pages=229−278 |isbn=978-1-4704-5263-6 |doi=10.1090/surv/238 |mr=3966406 |zbl=1422.14004 |s2cid=182109921 }}</ref> On the other hand, in the theta series of the four-dimensional [[F4 (mathematics)#F4 lattice|''body-centered cubic lattice'']] <math>\mathbb {F}_{4}</math> — whose geometry with <math>\mathbb {D}_{4}</math> defines [[Hurwitz quaternion]]s of even and odd square [[Norm (mathematics)|norm]] as realized in the <math>24</math>[[24-cell honeycomb|–cell honeycomb]] that is dual to the <math>16</math>–cell honeycomb (and, as a union of two self-dual tesseractic <math>8</math>[[Tesseractic honeycomb|–cell honeycombs]]) — the sixteenth <math>144 = 12^{2}</math> coefficient is the seven hundred and forty-fourth coefficient in the series;<ref name="F4Latt">{{Cite OEIS |A004013 |Theta series of body-centered cubic (b.c.c.) lattice. |access-date=2023-12-28 }}</ref> with coefficient index <math>144</math> the forty-ninth non-zero norm.<ref name="F4N">{{Cite OEIS |A004014 |Norms of vectors in the b.c.c. lattice. |access-date=2023-12-28 }}</ref>{{efn-lg|1={{math|1=<math>\mathbb{D}</math>{{su|b=4|p={{px2}}<sub>+</sub>}} [[Isomorphism|≅]] <math>\mathbb{Z}</math><sup>4</sup> }} as a lattice is the union of two {{math|1=<math>\mathbb{D}</math><sub>4</sub> }} lattices,<ref name="JiH" /> where the associated theta series of {{math|1=<math>\mathbb{Z}</math><sup>4</sup> }} has 744 as its '''50'''th indexed coefficient (as with theta series of {{math|1=<math>\mathbb{D}</math><sub>4</sub> }}),<ref name="ThZ4">{{Cite OEIS |A000118 |Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4. |access-date=2023-12-23 }}</ref><ref name="ThD4" /> and where its twenty-fifth coefficient is [[248 (number)|248]], which is the most important divisor of 744. Also, in this theta series of {{math|1=<math>\mathbb{Z}</math><sup>4</sup> }}, the preceding 49th coefficient is 456, the only number to hold an [[Aliquot sum|aliquot sum]] of 744,<ref name="AliSum" /> where {{math|1=744 − 456 = [[288 (number)|288]] }}, a value that is the number of cells in the disphenoidal 288-cell, whose 48 [[Vertex (geometry)|vertices]] collectively represent the twenty-four [[Hurwitz quaternion|Hurwitz unit quaternions]] with [[Norm (mathematics)|norm]] squared '''1''', united with the twenty-four vertices of the dual [[24-cell|24-cell]] with norm squared '''2'''. For the theta series of {{math|1=<math>\mathbb{D}</math><sub>4</sub> }}, that is realized in the [[16-cell honeycomb|'''16'''-cell honeycomb]], all {{math|1=25 × 2<sup>''m''</sup> }} indexed coefficients (i.e. 25, '''[[50 (number)|50]]''', 100, 200, 400, ...) are 744.<ref name="ThZ4" /> For both the theta series of {{math|1=<math>\mathbb{D}</math><sub>4</sub> }} and {{math|1=<math>\mathbb{Z}</math><sup>4</sup> }}, the 456th coefficient is 1920, the [[Divisor function#Definition|sum-of-divisors]] of 744.<ref name="SigmaSoD" /><br />Also, for the theta series of the four-dimensional {{math|1=<math>\mathbb{F}</math><sub>4</sub> }} lattice, coefficient index 288 is the 97th non-zero norm, coeff. index 456 the 153rd (an index that represents the seventeenth [[Triangular number|triangular number]]), and coeff. index 744 the '''250'''th;<ref name="F4N" /> the latter, a coefficient index that is the largest of only four numbers to hold a Euler totient of [[100|100]]: 101, 125, 202, and [[250 (number)|250]]<ref name="Etot" /> — the smallest of these is the twenty-sixth prime number, while the largest {{math|1=250 = 2 × 5<sup>3</sup> }} has a sum of prime factors that is [[17 (number)|17]], the seventh prime number;<ref name="PNum" /> and where {{math|1=202 = 49 + 153 }} and {{math|1=250 = 153 + 97 }}, with {{math|1=549 = 49 + 97 + 153 + 250 }}, the 447th indexed composite number.<ref name="CompNum" />{{efn-lr|1=More deeply, for the theta series of {{math|1=<math>\mathbb{F}</math><sub>4</sub> }} 744 is a prime-indexed coefficient over its first six indices less than {{math|1=10<sup>4</sup> }} (respectively, the 458th, 526th, 742nd, 799th, 842nd, and 1141st prime indices, with a sum of 4058; the latter in equivalence with {{math|1=1141 = [[7|7]] × [[163 (number)|163]] }}).<ref name="F4Latt" /> The first composite coeff. index in the series with coefficient 744 is its seventh index {{math|1=9251 = 11 × 29<sup>2</sup> }}, whose sum of prime factors, inclusive of 1, is [[70 (number)|70]] (which is of algebraic significance in terms of the construction of the twenty-four-dimensional [[Leech lattice]]). 456, and 1176, the [[Aliquot sum|aliquot sum]] of 744,<ref name="AliSum" /> are also prime-indexed coefficients over their first two coefficient indices (346th, 364th, and 1098th, 1159th, respectively).<ref name="F4Latt" /> }} }}

=== E<sub>8</sub> and the Leech lattice ===

Within finite simple groups [[Group of Lie type|of Lie type]], [[Exceptional Lie algebra|exceptional Lie algebra]] <math>\mathfrak{e_{8}}</math> holds a minimal [[Faithful representation|faithful representation]] in two hundred and forty-eight dimensions, where <math>248</math> divides <math>744</math> thrice over.<ref>{{Cite book |last=Pekonen |first=Osmo |chapter=Which Integer Is the Most Mysterious? |chapter-url=https://archive.bridgesmathart.org/2018/bridges2018-439.html#gsc.tab=0 |title=Bridges: Mathematics, Art, Music, Architecture, Education, Culture |series=Proceedings of the Bridges Conference |editor-last=Sarhangi |editor-first=Reza |year=2018 |page=441 |isbn=9781938664274 |issn=1590-5896 |s2cid=133028298 |oclc=7788577569 }}</ref><ref name="HeMcKay">{{Cite arXiv |last1=He |first1=Yang-Hui |last2=McKay |first2=John |title=Sporadic and Exceptional |year=2015 |pages=1–49 |class=math.AG |eprint=1505.06742 }} {{s2cid|117030188}}.
:"Of particular curiosity is the less well-known fact – in parallel to the above identity – that the constant term of the j-invariant, viz., 744, satisfies {{math|1=744 = 3 × 248. }} The number 248 is, of course, the dimension of the adjoint of the largest exceptional algebra [{{math|1=𝖊<sub>8</sub> }}]. In fact, that j should encode the presentations of [{{math|1=𝖊<sub>8</sub> }}] was settled long before the final proof of the Moonshine conjectures. This relationship between the largest sporadic group and the largest exceptional algebra would connect the McKay Correspondence to Moonshine and thereby weave another beautiful thread into the panoramic tapestry of mathematics."{{rp|'''p.4'''}}
:"<sup>'''16'''</sup> Incidentally, the reader is also alerted to the curiosity that {{math|1=σ<sub>1</sub>(240) = 744 }}."{{rp|'''p.21''' (note)}}</ref>{{rp|p.4}} [[Biography:John McKay (mathematician)|John McKay]] noted an intersection between finite simple groups of Lie type and those that are sporadic, where symmetries of nodes in the [[Dynkin diagram]]s of complex Lie algebra <math>\mathfrak{e}_{8}</math> as well as those of <math>\mathfrak{e}_{7}</math> and <math>\mathfrak{e}_{6}</math> respectively coincide with the three largest [[Conjugacy class|conjugacy class]]es of <math>\mathrm {F_{1}}</math>; where also the corresponding [[Monstrous moonshine|McKay–Thompson series]] <math>j(3\tau)^{1/3}</math> of sporadic [[Thompson sporadic group|Thompson group]] <math>\mathrm {Th}</math> holds coefficients representative of its faithful dimensional representation (also minimal at <math>\operatorname {dim} 248</math>)<ref>{{Cite OEIS |A007245 |McKay-Thompson series of class 3C for the Monster group. |access-date=2023-10-05 }}
:{{math|1='''''j'''''(𝜏)<sup>1/3</sup> = ''q''<sup> −1/3</sup> (1 + 248''q'' + 4124''q''<sup>2</sup> + 34752''q''<sup>3</sup> + ...) }}</ref><ref name="JanChr" /> whose values themselves embed irreducible representation of <math>\mathfrak{e_{8}}</math>.<ref name=TGann>{{Cite book |last=Gannon |first=Terry |title=Moonshine beyond the monster: The bridge connecting algebra, modular forms and physics |chapter=Introduction: glimpses of the theory beneath Monstrous Moonshine |chapter-url=https://assets.cambridge.org/97810094/01586/excerpt/9781009401586_excerpt.pdf |series=Cambridge Monographs on Mathematical Physics |publisher={{wipe|Cambridge University Press}} |location=Cambridge, MA |year=2006 |pages=1–15 |isbn=978-0-521-83531-2 |oclc=1374925688 |mr=2257727 |zbl=1146.11026 }}
:"In particular, {{math|1=4124 = 3875 + 248 + 1 }} and {{math|1=34752 = 30380 + 3875 + 2 · 248 + 1 }}, where 248, 3875 and 30380 are all dimensions of irreducible representations of {{math|1='''E'''<sub>8</sub>(<math>\mathbb{C}</math>) }}."{{rp|'''p.6'''}}</ref>{{rp|p.6}} In turn, exceptional Lie algebra <math>\mathfrak{e_{8}}</math> is shown to have a graded dimension <math>j(q)^{1/3}</math><ref>{{Cite OEIS |A121732 |Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order. |access-date=2023-10-05 }}</ref> whose [[Character table|character]] <math>\chi</math> lends to a [[Direct sum|direct sum]] equivalent to,<ref name=TGann />{{rp|p.7,9–11}}

:<math>\chi_{e_{8} \oplus e_{8} \oplus e_{8}} (q) = J(q) + 744 = j(q),</math> where the [[Conformal field theory|CFT]] probabilistic [[Partition function (mathematics)#Correlation functions|partition function]] for <math>\mathrm {F_{1}}</math> is <math>J(q)</math> of character <math>\chi_{F_{1}}.</math><ref name="GebMatt">{{Cite journal |last=Gaberdiel |first=Matthias R. |title=Constraints on extremal self-dual CFTs |journal=[[Physics:Journal of High Energy Physics|Journal of High Energy Physics]] |publisher=[[Company:Springer Science+Business Media|Springer]] |number=11, 087 |year=2007 |volume=2007 |pages=10–11 |bibcode=2007JHEP...11..087G |doi=10.1088/1126-6708/2007/11/087 |arxiv=0707.4073 |mr=2362062 |s2cid=16635058 |doi-access=free }}</ref>

The twenty-four dimensional [[Leech lattice]] <math>\Lambda_{24}</math> in turn can be constructed using three copies of the associated <math>\mathbb {E_{8}}</math> lattice<ref>{{Cite book |last1=Conway |first1=John H. |last2=Sloane |first2=N. J. A. |chapter-url=https://link.springer.com/chapter/10.1007/978-1-4757-2016-7_8  |title=Sphere Packings, Lattices and Groups |chapter=Algebraic Constructions for Lattices |publisher=Springer |location=New York, NY |year=1988 |isbn=978-1-4757-2016-7 |eissn=2196-9701  |doi=10.1007/978-1-4757-2016-7 |mr=1541550 }}</ref><ref name="HSMRP" />{{rp|pp.233–235}}{{efn-lg|1=The [[E8 lattice|<math>\mathbb{E}</math><sub>8</sub> lattice]] holds two hundred and forty root vectors that are represented by the [[Vertex arrangement|vertex arrangement]] of the {{math|1='''4'''<sub>21</sub> }} polytope, whose [[Petrie polygon]] is a thirty-sided [[Triacontagon|triacontagon]],<ref>{{Cite journal |last=Coxeter |first=H. S. M. |title=Seven Cubes and Ten 24-Cells |url=https://link.springer.com/content/pdf/10.1007/PL00009338.pdf |journal=Discrete & Computational Geometry |volume=19 |year=1998 |issue=2 |pages=156–157 |doi=10.1007/PL00009338 |doi-access=free |zbl=0898.52004 |s2cid=206861928 }}</ref><ref>{{Cite journal |last1=Coxeter |first1=H. S. M. |author-link1=Harold Scott MacDonald Coxeter |last2=Shephard |first2=G.C. |author2-link=Geoffrey Colin Shephard |title=Portraits of a Family of Complex Polytopes |journal=Leonardo |volume=25 |number=3/4 |year=1992 |pages=243–244 |doi=10.2307/1575843 |jstor=1575843 |s2cid=124245340 |zbl=0803.51023 }}</ref> where 30 is the Coxeter number {{math|1='''''h''''' }} of [[Coxeter group]] {{math|1='''E'''<sub>8</sub> }}. Also, Coxeter numbers of simple reflexions in {{math|'''E'''<sub>6</sub> }} and {{math|1='''E'''<sub>7</sub> }} are of orders 12 and 18, together equivalent to the Coxeter number of {{math|1='''E'''<sub>8</sub> }};<ref name="HSMRP" />{{rp|234}} where {{math|'''E'''<sub>6</sub> }} and {{math|1='''E'''<sub>7</sub> }} ''embed'' inside {{math|1='''E'''<sub>8</sub> }}. Four-dimensional {{math|1='''H'''<sub>4</sub> }} [[H4 polytope|hexadecachoric]] symmetry also contains a Coxeter number of [[30 (number)|30]], where {{math|1='''H'''<sub>4</sub> }} is the higher-dimensional analogue of icosahedral symmetry {{math|1='''I'''<sub>h</sub> }}.<br />This <math>\mathbb{E}</math><sub>8</sub> lattice structure with '''240''' root vectors can be constructed with '''120''' quaternionic [[Icosian|unit icosians]] that form the vertices of the four-dimensional [[600-cell|600-cell]], whose symmetries are rooted in three-dimensional [[Icosahedral symmetry|icosahedral symmetry]] {{math|1='''I'''<sub>h</sub> }} of order 120,<ref>{{Cite journal |last=Baez |first=John C. |author-link=John C. Baez |title=From the Icosahedron to E<sub>8</sub> |journal=London Mathematical Society Newsletter |volume=476 |pages=18–23 |year=2018 |arxiv=1712.06436 |mr=3792329 |s2cid=119151549 |zbl=1476.51020 }}</ref> a value equal to the total number of reflections of [[Coxeter group]] {{math|1='''E'''<sub>8</sub> }}.<ref name="HSMRP">{{Cite book |last=Coxeter |first=H. S. M. |author-link=Harold Scott MacDonald Coxeter |title=Regular Polytopes |publisher=Methuen & Co. |edition=1st |year=1948 |location=London |pages=1–321 |isbn=9780521201254 |oclc=472190910 |mr=0027148 }}</ref>{{rp|226–232}} In total, a [[Regular icosahedron|regular icosahedron]] and [[Regular dodecahedron|dodecahedron]] contain thirty-one axes of symmetry; six five-fold, ten three-fold, and fifteen two-fold.<ref>{{Cite book |editor-last=Sarhangi |editor-first=Reza |author-last=Hart |author-first=George W. |author-link= George W. Hart |chapter=Icosahedral Constructions |chapter-url=http://t.archive.bridgesmathart.org/1998/bridges1998-195.pdf |title=Bridges: Mathematical Connections in Art, Music, and Science |series=Proceedings of the Bridges Conference |location=Winfield, Kansas |year=1998 |page=196 |isbn=978-0966520101 |oclc=59580549 |s2cid=202679388 }}</ref> <br />Where the arithmetic mean of divisors of 744 is 120,<ref name="ArithNum0" /><ref name="ArithNum" /> with a largest ''prime factor'' of '''31''' and a reduced totient {{math|1='''λ'''(''n'') }} of '''30''',<ref name="redTot" /> the number of integers that are relatively prime with and up to 744 is 240,<ref name="Etot" /> with the sum-of-divisors {{math|1=σ(240) = 744 }}.<ref name="HeMcKay" />{{rp|p.21}} }} and with the eight-dimensional [[Octonion|octonion]]s <math>\mathbb {O}</math> (see also, [[Freudenthal magic square]]),<ref>{{Cite journal |last1=Wilson |first1=Robert A. |author-link=Robert Arnott Wilson |title=Octonions and the Leech lattice |mr=2542837 |year=2009 |journal=Journal of Algebra |volume=322 |issue=6 |pages=2186–2190|doi=10.1016/j.jalgebra.2009.03.021 |doi-access=free }}</ref> where the automorphism group of <math>\mathbb {O}</math> is the smallest exceptional Lie algebra <math>\mathfrak{g_{2}}</math>, which embeds inside <math>\mathfrak{e_{8}}</math>. In the form of a [[Vertex operator algebra|vertex operator algebra]], the Leech lattice '''VOA''' is the first aside from <math>V_1</math> (as <math>\mathbb {C}_{24}</math>) with a [[Physics:Central charge|central charge]] <math>c</math> of <math>24</math>, out of a total seventy-one such modular invariant [[Conformal field theory|conformal field theories]] of holomorphic VOAs of weight one.<ref>{{Cite journal |last1=Schellekens |first1=Adrian Norbert |title=Meromorphic ''c'' = 24 conformal field theories. |url=https://link.springer.com/article/10.1007/BF02099044 |journal=[[Physics:Communications in Mathematical Physics|Communications in Mathematical Physics]] |publisher=[[Company:Springer Science+Business Media|Springer]] |volume=153 |issue=1 |year=1993 |pages=159–185 |location=Berlin |doi=10.1007/BF02099044 |arxiv=hep-th/9205072 |bibcode=1993CMaPh.153..159S |mr=1213740 |zbl=0782.17014 |s2cid=250425623 }}</ref> Known as '''''Schellekens' list''''', these algebras form [[Leech lattice#Geometry|deep hole]]s in <math>V_{\Lambda_{24}}</math> whose corresponding [[Orbifold|orbifold construction]]s are [[Isomorphism|isomorphic]] to the moonshine module <math>V_{2}</math><sup>{{large|♮}}</sup> that contains <math>\mathrm {F_{1}}</math> as its automorphism;<ref>{{Cite journal |last1=Möller |first1=Sven |last2=Scheithauer |first2=Nils R. |title=Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra. |url=https://annals.math.princeton.edu/2023/197-1/p04 |journal=[[Annals of Mathematics]] |volume=197 |issue=1 |publisher=[[Organization:Princeton University|Princeton University]] & the Institute for Advanced Study |year=2023 |pages=261–285 |doi=10.4007/annals.2023.197.1.4 |bibcode=2019arXiv191004947M |mr=4513145 |zbl=07623024 |arxiv=1910.04947 |s2cid=204401905 }}</ref> of these, the second and third largest contain [[Affine group|affine structures]] <math>E^3_{8,1}</math> and <math>D_{16,1}E_{8,1}</math> that are realized in <math>\operatorname {dim}744</math>.{{efn-lg|1={{math|1=''V''{{sub|''E''{{su|b=8|p={{px2}}<sub>3</sub> }} }} }} is isomorphic to the [[Tensor product|tensor product]] {{math|1=''V''{{sub|''E''<sub>8</sub> }} {{resize|70%|⊗ }} ''V''{{sub|''E''<sub>8</sub> }} {{resize|70%|⊗ }} ''V''{{sub|''E''<sub>8</sub> }} }}; also, affine structure {{math|1=''D''<sub>16</sub> }} is different from {{math|1=''D''{{su|b=16|p={{px2}}<sub>+</sub>}} }}, which is associated with even positive definite [[Unimodular lattice|unimodular lattice]] {{math|1=<math>\mathbb{D}</math>{{su|b=16|p={{px2}}<sub>+</sub>}} }}.<ref>{{Cite journal |last1=Griess, Jr. |first1=Robert L. |last2=Lam |first2=Ching Hung |url=http://www.math.lsa.umich.edu/~rlg/researchandpublications/pdffiles/moonshinepath13oct09.pdf |title=A moonshine path from E8 to the Monster |journal=[[Journal of Pure and Applied Algebra]] |volume=215 |issue=5 |year=2011 |pages=930–931 |doi=10.1016/j.jpaa.2010.07.001 |doi-access=free |mr=2747229 |s2cid=123613651 |zbl=1213.17028 }}</ref><ref>{{Cite journal |last1=Griess Jr. |first1=Robert L. |author1-link=Robert Griess |last2=Lam |first2=Ching Hung |title=A moonshine path for 5 A and associated lattices of ranks 8 and 16 |url=https://www.sciencedirect.com/science/article/pii/S002186931000623X |journal=[[Journal of Algebra]] |volume=331 |publisher=[[Company:Elsevier|Elsevier]] |location=Amsterdam |year=2011 |page=348 |doi=10.1016/j.jalgebra.2010.11.013 |bibcode=2010arXiv1006.3907G |mr=2774662 |zbl=1283.20009 |arxiv=1006.3907 |s2cid=119679255 }}</ref><br /> {{math|1=''E''{{su|b=8,1|p={{px2}}<sub>3</sub> }} }} and {{math|''D''<sub>16,1</sub>''E''<sub>8,1</sub> }} are associated with codes {{math|1=''e''{{su|b=8|p={{px2}}<sub>3</sub> }} }} and {{math|''d''<sub>16</sub>''e''<sub>8</sub> }} that are two of only nine in-equivalent doubly even self-dual codes of length 24 and  weight 4.<ref>{{Cite journal |last1=Harada |first1=Masaaki |last2=Lam |first2=Ching Hung |last3=Munemasa |first3=Akihiro |title=Residue codes of extremal Type II Z<sub>4</sub>-codes and the moonshine vertex operator algebra. |url=https://link.springer.com/article/10.1007/s00209-012-1091-z |journal=[[Mathematische Zeitschrift]] |volume=274 |issue=1 |pages=691, 698–699 |publisher=[[Company:Springer Science+Business Media|Springer]] |year=2013 |doi=10.1007/s00209-012-1091-z |bibcode=2010arXiv1005.1144H |arxiv=1005.1144 |mr=3054350 |zbl=1283.94112 |s2cid=253730965 }}</ref><ref>{{Cite journal |last1=Pless |first1=V. |last2=Sloane |first2=N. J. A. |author2-link=Neil Sloane |title=On the classification and enumeration of self-dual codes. |journal=Journal of Combinatorial Theory | series=Series A |volume=18 |publisher=[[Company:Elsevier|Elsevier]] |location=Amsterdam |year=1975 |issue=3 |pages=313–335 |doi=10.1016/0097-3165(75)90042-4 |doi-access=free |mr=0376232 |zbl=0305.94011 |s2cid=16303075 }}</ref> The largest of these VOAs {{math|1=''V''{{sub|''D''<sub>24</sub> }} }} is realized in {{math|1=dim 1128 }}, where successively halving its dimensional space leads to a {{frac|70|1|2}}–dimensional space. }}<ref>{{Cite journal |last1=Van Ekeren |first1=Jethro |last2=Lam |first2=Ching Hung |last3=Möller |first3=Sven |last4=Shimakura |first4=Hiroki |title=Schellekens' list and the very strange formula |url=https://www.sciencedirect.com/science/article/abs/pii/S0001870821000050 |journal=[[Advances in Mathematics]] |volume=380 |publisher=[[Company:Elsevier|Elsevier]] |location=Amsterdam |year=2021 |pages=1–34 (107567) |doi=10.1016/j.aim.2021.107567 |mr=4200469 |zbl=1492.17027 |s2cid=218870375 |arxiv=2005.12248 }}</ref><ref>{{Cite journal |last1=Betsumiya |first1=Koichi |last2=Lam |first2=Ching Hung |last3=Shimakura |first3=Hiroki |title=Automorphism Groups and Uniqueness of Holomorphic Vertex Operator Algebras of Central Charge 24 |url=https://link.springer.com/article/10.1007/s00220-022-04585-6 |journal=[[Physics:Communications in Mathematical Physics|Communications in Mathematical Physics]] |volume=399 |issue=3 |publisher=[[Company:Springer Science+Business Media|Springer]] |location=Berlin |year=2023 |pages=1773–1810 |doi=10.1007/s00220-022-04585-6 |bibcode=2023CMaPh.399.1773B |mr=4580534 |arxiv=2203.15992 |zbl=07681021 |s2cid=247793117 }}</ref>

== Other properties ==

744 is also the sum of consecutive pentagonal numbers <math>P_{n}</math>,<ref>{{Cite OEIS |A129863 |Sums of three consecutive pentagonal numbers. |access-date=2023-07-16 }}</ref><ref>{{Cite OEIS |A000326 |Pentagonal numbers. |access-date=2023-07-16 }}</ref>{{efn-lr|1=Where the smallest non-unitary pentagonal pyramidal number is [[6|6]], the eleventh is 726 = 6! + 6, and the twenty-fourth is [[7000 (number)#7200|7200]],<ref>{{Cite OEIS |A002411 |Pentagonal pyramidal numbers: a(n) equal to n^2*(n+1)/2. |access-date=2023-11-09 }}</ref> which is a number with a Euler totient value of 1920,<ref name="Etot" /> and a reduced totient of 120.<ref name="redTot" /> }}
<math display=block>744 = P_{11} + P_{12} + P_{13} = 210 + 247 + 287. </math>

It is also the [[Magic constant|magic constant]] of a six by six [[Magic square|magic square]] consisting of thirty-six consecutive prime numbers, between [[41 (number)|41]] and [[223 (number)|223]] inclusive.<ref>{{Cite OEIS |A177434 |The magic constants of 6 X 6 magic squares composed of consecutive primes. |access-date=2023-07-16 }}</ref>{{efn-lr|content=The magic square is:
<br />
<math>\begin{bmatrix}
  139 & 113 & 151 & 131 & 83 & 127 \\
 223 & 149 & 89 & 47 & 157 & 79 \\
 173 & 103 & 181 & 167 & 59 & 61 \\
 67 & 137 & 53 & 97 & 211 & 179 \\
 101 & 199 & 73 & 109 & 71 & 191 \\
 41 & 43 & 197 & 193 & 163 & 107 
\end{bmatrix}</math>
<br />
This is the second-smallest magic constant for a {{math|1=6 '''×''' 6 }} magic square consisting of thirty-six consecutive prime numbers, where the sum between the smallest and largest prime numbers in this square is equal to {{math|1=41 + 223 = 264 }}. The smallest such constant is {{math|1=484 = 22<sup>2</sup> }}<ref name="PrimeMS">{{Cite OEIS |A177434 |The magic constants of 6 X 6 magic squares composed of consecutive primes. |access-date=2023-09-11 }}</ref> whose aliquot sum of 447 is the reverse permutation of the digits of 744 in decimal;<ref name="AliSum" /> specifically, [[22 (number)|22]] and 264 are respectively the twelfth and fourteenth numbers {{math|1=''n'' }} whose squares are [[Undulating number|''undulating'']] in decimal, while the thirteenth and penultimate such known number is '''26'''.<ref>{{Cite OEIS |A122875 |Numbers whose square is undulating. |access-date=2023-11-02 }}</ref><ref>{{Cite OEIS |A016073 |Undulating squares. |access-date=2023-11-02 }}</ref><ref>{{Cite book |last=Pickover |first=Clifford A. |chapter=Chapter 20: The Undulation of the Monks |title=Keys to Infinity |publisher=[[Company:Wiley (publisher)|John Wiley & Sons, Inc.]] |location=New York, NY |year=1995 |pages=xviii; 1−332 |isbn=978-0471193340 |oclc=31710129 |s2cid=121761396 }}</ref>{{rp|pp.159,160}} The smallest possible magic constant of an {{math|1=''n'' × ''n'' }} magic square consisting only of ''distinct'' prime numbers is 120, from an {{math|1=''n'' }} of {{math|1=4 }},<ref>{{Cite OEIS |A164843 |The smallest magic constant of an n X n magic square with distinct prime entries. |access-date=2023-11-05 }}</ref><ref>{{Cite web |last= Makarova |first=Natalia V. |title=Least Magic Squares of Prime Numbers (Part I) |website=klassikpoez.narod.ru |url=http://www.natalimak1.narod.ru/sqmin1.htm |language=ru |archive-url=http://archive.today/UsMgq |archive-date=2023-12-07 }} Archived translation in English. Retrieved 2023-12-07.</ref> a value equal to the arithmetic mean of all sixteen divisors of 744.<ref name="ArithNum0" /><ref name="ArithNum" /> <br />An {{math|1=11 '''×''' 11 }} magic square that is normal has a magic constant of 671,<ref>{{Cite OEIS |A006003 |a(n) equal to n*(n^2 + 1)/2 |access-date=2023-09-11 }}</ref> which is the sixth and largest composite number to have a sum-of-divisors equal to 744.<ref name="SigmaSoD" /> }}

744 is the number of non-congruent [[Polygon partition|polygonal regions]] in a regular '''36'''-[[Polygon|gon]] with all diagonals drawn.<ref>{{Cite OEIS |A187781 |Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn. |access-date=2023-12-23 }}</ref>{{efn-lr|1=Otherwise, {{math|1='''992''' = 248 + 744 }} is the least possible number of [[Diagonal|diagonal]]s of a simple convex [[Polyhedron|polyhedron]] with thirty-six [[Face (geometry)#Face or 2-face|faces]]; for sixteen and twenty faces, there are respectively a minimum of 132 and 240 diagonals<ref>{{Cite OEIS |A279019 |Least possible number of diagonals of simple convex polyhedron with n faces. |access-date=2024-01-10 }}</ref> (values which represent the prime index of the largest prime totative of 744 and the count of all its totatives),<ref name="Etot" /> where {{math|1=132 + 240 = 372 }}, or half 744 (alongside respective indices that generate a sum of 36). }}

There are 744 ways in-which fourteen [[Square|square]]s of different sizes fit [[Rectangle#Tessellations|edge-to-edge]] inside a larger [[Rectangle|rectangle]].<ref>{{Cite OEIS |A002839 |Number of simple perfect squared rectangles of order n up to symmetry. |access-date=2023-07-16 }}</ref>{{efn-lr|1=On the other hand, '''456''' (the only number to have an aliquot sum of 744)<ref name="AliSum" /> is the maximum number of unit squares — only joined at corners — that can be inscribed inside a <math> 40 \times 40</math> square (of area '''1600''').<ref>{{Cite OEIS |A326118 |a(n) is the largest number of squares of unit area connected only at corners and without holes that can be inscribed in an n X n square. |access-date=2024-01-20 }}</ref> }}

== See also ==
*Moonshine theory – the study between the relationships held by F<sub>1</sub> and modular functions, in particular ''j(τ)''
:*[[Heegner number]]s — [[1|1]], [[2|2]], [[3|3]], [[7|7]], [[11 (number)|11]], [[19 (number)|19]], [[43 (number)|43]], [[67 (number)|67]], and [[163 (number)|163]]
:*[[Supersingular prime (moonshine theory)|Supersingular primes]] — 2, 3, 5, 7, [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], and [[31 (number)|31]], as well as [[41 (number)|41]], [[47 (number)|47]], [[59 (number)|59]], and [[71 (number)|71]]
:*The [[J-invariant|j-invariant]] — [[1728 (number)|1728]] = [[12 (number)|12]]<sup>3</sup> = ''j(i)'' , one less than the first [[Taxicab number|taxicab number]] [[1729 (number)|1729]] (also known as the ''Ramanujan–Hardy number'')

*[[Digit-reassembly number|Osiris]] or digit-reassembly numbers – equal to the sum of all permutations of subsets of their own digits (e.g. [[132 (number)|132]] in decimal vis-à-vis [[Symmetric group|symmetric group]] [[Steiner system#Construction from K6 graph factorization|''S<sub>6</sub>'']])

== Notes ==
;Higher arithmetic
{{Notelist-lr|40em}}

;Heegner numbers, finite simple groups, and lattices D<sub>4</sub>, F<sub>4</sub>, E<sub>8</sub>, Λ<sub>24</sub>
{{Notelist-lg}}

== References ==
{{Reflist}}
<br />
{{Number theory}}

{{DEFAULTSORT:743 (Number)}}
[[Category:Integers]]

{{Sourceattribution|744 (number)}}