27 (number)

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Short description: Natural number
← 26 27 28 →
Cardinaltwenty-seven
Ordinal27th
Factorization33
Divisors1, 3, 9, 27
Greek numeralΚΖ´
Roman numeralXXVII
Binary110112
Ternary10003
Quaternary1234
Quinary1025
Senary436
Octal338
Duodecimal2312
Hexadecimal1B16
Vigesimal1720
Base 36R36

27 (twenty-seven; Roman numeral XXVII) is the natural number following 26 and preceding 28.

Mathematics

Twenty-seven is the cube of 3, or three tetrated [math]\displaystyle{ ^{2}3 = 3^{3} = 3\times 3\times 3 }[/math], divisible by the number of prime numbers below it (nine).

The first non-trivial decagonal number is 27.[1]

27 has an aliquot sum of 13[2] (the sixth prime number) in the aliquot sequence [math]\displaystyle{ (27, 13, 1, 0) }[/math] of only one composite number, rooted in the 13-aliquot tree.[3]

The sum of the first four composite numbers is [math]\displaystyle{ 4 + 6 + 8 + 9 = 27 }[/math],[4] while the sum of the first four prime numbers is [math]\displaystyle{ 2 + 3 + 5 + 7 = 17 }[/math],[5] with 7 the fourth indexed prime.[6][lower-alpha 1]

In the Collatz conjecture (i.e. the [math]\displaystyle{ 3n+1 }[/math] problem), a starting value of 27 requires 3 × 37 = 111 steps to reach 1, more than any smaller number.[10][lower-alpha 2]

27 is also the fourth perfect totient number — as are all powers of 3 — with its adjacent members 15 and 39 adding to twice 27.[13][lower-alpha 3]

A prime reciprocal magic square based on multiples of [math]\displaystyle{ \tfrac {1}{7} }[/math] in a [math]\displaystyle{ 6 \times 6 }[/math] square has a magic constant of 27.

Including the null-motif, there are 27 distinct hypergraph motifs.[14]

The Clebsch surface, with 27 straight lines

There are exactly twenty-seven straight lines on a smooth cubic surface,[15] which give a basis of the fundamental representation of Lie algebra [math]\displaystyle{ \mathrm {E_{6}} }[/math].[16][17]

The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional;[18] its automorphism group is the 52-dimensional exceptional Lie algebra [math]\displaystyle{ \mathrm {F_{4}}. }[/math][19]

There are twenty-seven sporadic groups, if the non-strict group of Lie type [math]\displaystyle{ \mathrm {T} }[/math] (with an irreducible representation that is twice that of [math]\displaystyle{ \mathrm {F_{4}} }[/math] in 104 dimensions)[20] is included.[21]

In Robin's theorem for the Riemann hypothesis, twenty-seven integers fail to hold [math]\displaystyle{ \sigma(n) \lt e^\gamma n \log \log n }[/math] for values [math]\displaystyle{ n \leq 5040, }[/math] where [math]\displaystyle{ \gamma }[/math] is the Euler–Mascheroni constant; this hypothesis is true if and only if this inequality holds for every larger [math]\displaystyle{ n. }[/math][22][23][24]

Base-specific

In decimal, 27 is the first composite number not divisible by any of its digits, as well as:

  • the third Smith number[25] and sixteenth Harshad number,[26]
  • the only positive integer that is three times the sum of its digits,
  • equal to the sum of the numbers between and including its digits: [math]\displaystyle{ 2+3+4+5+6+7=27 }[/math].

Also in base ten, if one cyclically rotates the digits of a three-digit number that is a multiple of 27, the new number is also a multiple of 27. For example, 378, 783, and 837 are all divisible by 27.

  • In similar fashion, any multiple of 27 can be mirrored and spaced with a zero each for another multiple of 27 (i.e. 27 and 702, 54 and 405, and 378 and 80703 are all multiples of 27).
  • Any multiple of 27 with "000" or "999" inserted yields another multiple of 27 (20007, 29997, 50004, and 59994 are all multiples of 27).

In senary (base six), one can readily test for divisibility by 43 (decimal 27) by seeing if the last three digits of the number match 000, 043, 130, 213, 300, 343, 430, or 513.

In decimal representation, 27 is located at the twenty-eighth (and twenty-ninth) digit after the decimal point in π:

[math]\displaystyle{ 3.141\;592\;653\;589\;793\;238\;462\;643\;383\;{\color{red}27}9\ldots }[/math]

If one starts counting with zero, 27 is the second self-locating string after 6, of only a few known.[27][28]

In science

Astronomy

Electronics

  • The type 27 vacuum tube (valve), a triode introduced in 1927, was the first tube mass-produced for commercial use to incorporate an indirectly heated cathode. This made it the first vacuum tube that could function as a detector in AC-powered radios. Prior to the introduction of the 27, home radios were powered by a set of three or more storage batteries with voltages of 3 volts to 135 volts.

In language and literature

  • The number of letters in the Spanish alphabet.[33]
  • The number of books in the New Testament.
  • The total number of letters in the Hebrew alphabet (22 regular letters and 5 final consonants).
  • Alternate name for The Hunt, a book by William Diehl.
  • Abbé Faria's prisoner number in the book The Count of Monte Cristo.
  • In Stephen King's novel It, It returns every 27 years to Derry.

In astrology

  • 27 Nakṣatra or lunar mansions in Hindu astrology.

In art

Movies

  • Summer, or 27 Missing Kisses
  • Chapter 27-I
  • 27 Dresses
  • Number 27 (written by Michael Palin)
  • The 27th Day (Science Fiction Film from 1957)

Music

  • "27", a song by Fall Out Boy on the album Folie à Deux
  • "27", a song by Passenger on the album Whispers
  • "27", a song by Title Fight on the album Shed.[34]
  • "27", a song by Biffy Clyro on the album Blackened Sky
  • "27", a song on Machine Gun Kelly's album Bloom
  • "27 Jennifers", a song by Mike Doughty on the album Rockity Roll
  • 27, an album by South Korean singer Kim Sung-kyu
  • 27, an album by Argentine rock band Ciro y los Persas
  • Twenty-Seven, an album by The Adicts.
  • French rapper Kaaris' signature number is 27, from his zip code 93270.
  • "Twenty Seven Strangers" by Villagers.
  • 27, a Boston-based band
  • 27, an opera by composer Ricky Ian Gordon and librettist Royce Vavrek
  • 27 Club, a list of popular musicians, artists, or actors who died at age 27
  • "Weird Al" Yankovic has a recurring joke involving the number 27, which is used in several songs.

Other

  • The Minneapolis-based artist Deuce 7 (a.k.a. Deuce Seven, Twenty Seven, 27).

In sports

  • The value of all the colors in snooker add up to 27.
  • The number of outs in a regulation baseball game for each team at all adult levels, including professional play, is 27.
  • The New York Yankees have won 27 World Series championships, the most of any team in the MLB.

In other fields

Twenty-seven is also:

  • A-27, American attack aircraft.
  • The code for international direct-dial phone calls to South Africa .
  • The name of a cigarette, Marlboro Blend No. 27.
  • The number of the French department Eure.

See also

  • List of highways numbered 27
  • List of highways numbered 27A

Notes

  1. ↑ Whereas the composite index of 27 is 17[7] (the cousin prime to 13),[8] 7 is the prime index of 17.[6]
    The sum  27 + 17 + 7 = 53  represents the sixteenth indexed prime (where 42 = 16).
    While 7 is the fourth prime number, the fourth composite number is 9 = 32, that is also the composite index of 16.[9]
  2. ↑ On the other hand,
    • The reduced Collatz sequence of 27, that counts the number of prime numbers in its trajectory, is 41.[11]
      This count represents the thirteenth prime number, that is also in equivalence with the sum of members in the aliquot tree (27, 13, 1, 0).[3][2]
    • The next two larger numbers in the Collatz conjecture to require more than 111 steps to return to 1 are 54 and 55
    • Specifically, the fourteenth prime number 43 requires twenty-seven steps to reach 1.
    The sixth pair of twin primes is (41, 43),[12] whose respective prime indices generate a sum of 27.
  3. ↑ Also,  36 = 62  is the sum between PTNs  39 – 15 = 24  and  3 + 9 = 12. In this sequence, 111 is the seventh PTN.

References

  1. ↑ "Sloane's A001107 : 10-gonal (or decagonal) numbers". OEIS Foundation. https://oeis.org/A001107. 
  2. ↑ 2.0 2.1 Sloane, N. J. A., ed. "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". OEIS Foundation. https://oeis.org/A001065. Retrieved 2023-10-31. 
  3. ↑ 3.0 3.1 Sloane, N. J. A., ed (January 11, 1975). "Aliquot sequences". Mathematics of Computation (OEIS Foundation) 29 (129): 101–107. https://oeis.org/wiki/Aliquot_sequences. Retrieved 2023-10-31. 
  4. ↑ Sloane, N. J. A., ed. "Sequence A151742 (Composite numbers which are the sum of four consecutive composite numbers.)". OEIS Foundation. https://oeis.org/A151742. Retrieved 2023-11-02. 
  5. ↑ Sloane, N. J. A., ed. "Sequence A007504 (Sum of the first n primes.)". OEIS Foundation. https://oeis.org/A007504. Retrieved 2023-11-02. 
  6. ↑ 6.0 6.1 Sloane, N. J. A., ed. "Sequence A000040 (The prime numbers.)". OEIS Foundation. https://oeis.org/A000040. Retrieved 2023-10-31. 
  7. ↑ Sloane, N. J. A., ed. "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". OEIS Foundation. https://oeis.org/A002808. Retrieved 2023-10-31. 
  8. ↑ Sloane, N. J. A., ed. "Sequence A046132 (Larger member p+4 of cousin primes (p, p+4).)". OEIS Foundation. https://oeis.org/A046132. Retrieved 2023-10-31. 
  9. ↑ Sloane, N. J. A., ed. "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". OEIS Foundation. https://oeis.org/A002808. Retrieved 2023-11-08. 
  10. ↑ Sloane, N. J. A., ed. "Sequence A112695 (Number of steps needed to reach 4,2,1 in Collatz' 3*n+1 conjecture.)". OEIS Foundation. https://oeis.org/A112695. Retrieved 2023-10-31. 
  11. ↑ Sloane, N. J. A., ed. "Sequence A286380 (a(n) is the minimum number of iterations of the reduced Collatz function R required to yield 1. The function R (A139391) is defined as R(k) equal to (3k+1)/2^r, with r as large as possible.)". OEIS Foundation. https://oeis.org/A286380. Retrieved 2023-11-08. 
  12. ↑ Sloane, N. J. A., ed. "Sequence A077800 (List of twin primes {p, p+2}.)". OEIS Foundation. https://oeis.org/A077800. Retrieved 2023-11-08. 
  13. ↑ Sloane, N. J. A., ed. "Sequence A082897 (Perfect totient numbers.)". OEIS Foundation. https://oeis.org/A082897. Retrieved 2023-11-02. 
  14. ↑ Lee, Geon; Ko, Jihoon; Shin, Kijung (2020). "Hypergraph Motifs: Concepts, Algorithms, and Discoveries". in Balazinska, Magdalena; Zhou, Xiaofang. 46th International Conference on Very Large Data Bases. Proceedings of the VLDB Endowment. 13. ACM Digital Library. pp. 2256–2269. doi:10.14778/3407790.3407823. ISBN 9781713816126. OCLC 1246551346. http://www.vldb.org/pvldb/volumes/13/. 
  15. ↑ Baez, John Carlos (February 15, 2016). "27 Lines on a Cubic Surface". American Mathematical Society. https://blogs.ams.org/visualinsight/2016/02/15/27-lines-on-a-cubic-surface/. 
  16. ↑ Aschbacher, Michael (1987). "The 27-dimensional module for E6. I". Inventiones Mathematicae (Heidelberg, DE: Springer) 89: 166–172. doi:10.1007/BF01404676. Bibcode: 1987InMat..89..159A. 
  17. ↑ Sloane, N. J. A., ed. "Sequence A121737 (Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order.)". OEIS Foundation. https://oeis.org/A121737. Retrieved 2023-10-31. 
  18. ↑ Kac, Victor Grigorievich (1977). "Classification of Simple Z-Graded Lie Superalgebras and Simple Jordan Superalgebras". Communications in Algebra (Taylor & Francis) 5 (13): 1380. doi:10.1080/00927877708822224. 
  19. ↑ Baez, John Carlos (2002). "The Octonions". Bulletin of the American Mathematical Society (Providence, RI: American Mathematical Society) 39 (2): 189–191. doi:10.1090/S0273-0979-01-00934-X. https://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/. 
  20. ↑ Lubeck, Frank (2001). "Smallest degrees of representations of exceptional groups of Lie type". Communications in Algebra (Philadelphia, PA: Taylor & Francis) 29 (5): 2151. doi:10.1081/AGB-100002175. https://www.tandfonline.com/doi/abs/10.1081/AGB-100002175. 
  21. ↑ Hartley, Michael I.; Hulpke, Alexander (2010). "Polytopes Derived from Sporadic Simple Groups". Contributions to Discrete Mathematics (Alberta, CA: University of Calgary Department of Mathematics and Statistics) 5 (2): 27. doi:10.11575/cdm.v5i2.61945. ISSN 1715-0868. https://cdm.ucalgary.ca/article/view/61945/46662. 
  22. ↑ Axler, Christian (2023). "On Robin's inequality". The Ramanujan Journal (Heidelberg, GE: Springer) 61 (3): 909–919. doi:10.1007/s11139-022-00683-0. Bibcode: 2021arXiv211013478A. 
  23. ↑ Robin, Guy (1984). "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann" (in French). Journal de Mathématiques Pures et Appliquées. Neuvième Série 63 (2): 187–213. ISSN 0021-7824. http://zakuski.utsa.edu/~jagy/Robin_1984.pdf. 
  24. ↑ Sloane, N. J. A., ed. "Sequence A067698 (Positive integers such that sigma(n) is greater than or equal to exp(gamma) * n * log(log(n)).)". OEIS Foundation. https://oeis.org/A067698. Retrieved 2023-10-31. 
  25. ↑ "Sloane's A006753 : Smith numbers". OEIS Foundation. https://oeis.org/A006753. 
  26. ↑ "Sloane's A005349 : Niven (or Harshad) numbers". OEIS Foundation. https://oeis.org/A005349. 
  27. ↑ Dave Andersen. "The Pi-Search Page". https://www.angio.net/pi/. 
  28. ↑ Sloane, N. J. A., ed. "Sequence A064810 (Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where 1 is the 0th digit.)". OEIS Foundation. https://oeis.org/A064810. Retrieved 2023-10-31. 
  29. ↑ "Dark Energy, Dark Matter | Science Mission Directorate". https://science.nasa.gov/astrophysics/focus-areas/what-is-dark-energy. 
  30. ↑ Steve Jenkins, Bones (2010), ISBN:978-0-545-04651-0
  31. ↑ "Catalog of Solar Eclipses of Saros 27". NASA. https://eclipse.gsfc.nasa.gov/SEsaros/SEsaros027.html. 
  32. ↑ "Catalog of Lunar Eclipses in Saros 27". NASA. https://eclipse.gsfc.nasa.gov/LEsaros/LEsaros027.html. 
  33. ↑ "SpanishDict Grammar Guide" (in en). https://www.spanishdict.com/guide/the-spanish-alphabet. 
  34. ↑ "Shed, by Title Fight". http://titlefightmusic.bandcamp.com/album/shed. 

Further reading

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987), p. 106.

External links



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