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{{short description|Class of natural numbers}}
In [[Mathematics|mathematics]], a '''superabundant number''' is a certain kind of [[Natural number|natural number]]. A natural number {{mvar|n}} is called ''superabundant'' precisely when, for all {{math|''m'' &lt; ''n''}}:

:<math>\frac{\sigma(m)}{m} < \frac{\sigma(n)}{n}</math>

where {{mvar|σ}} denotes the [[Divisor function|sum-of-divisors function]] (i.e., [[Aliquot sum|the sum]] of all positive [[Divisor|divisor]]s of {{mvar|n}}, including {{mvar|n}} itself). The first few superabundant numbers are {{nowrap|1, 2, 4, 6, [[12 (number)|12]], [[24 (number)|24]], [[36 (number)|36]], [[48 (number)|48]], [[60 (number)|60]], [[120 (number)|120]], ...}} {{OEIS|id=A004394}}. For example, the number [[5|5]] is not a superabundant number because for {{nowrap|1, 2, 3, 4}}, and 5, the sigma is {{nowrap|1, 3, 4, 7, 6}}, and {{nowrap|7/4 > 6/5}}.

Superabundant numbers were defined by {{harvs|first1=Leonidas|last1=Alaoglu|first2=Paul|last2=Erdős|year=1944|txt=yes}}. Unknown to Alaoglu and Erdős, about 30 pages of Ramanujan's 1915 paper "Highly Composite Numbers" were suppressed.  Those pages were finally published in ''[[The Ramanujan Journal]]'' 1 (1997), 119–153.  In section 59 of that paper, Ramanujan defines generalized [[Highly composite number|highly composite number]]s, which include the superabundant numbers.

== Properties ==
{{Euler_diagram_numbers_with_many_divisors.svg}}
{{harvs|first1=Leonidas|last1=Alaoglu|first2=Paul|last2=Erdős|year=1944|txt=yes}} proved that if ''n'' is superabundant, then there exist a ''k'' and ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''k''</sub> such that

:<math>n=\prod_{i=1}^k (p_i)^{a_i}</math>

where ''p''<sub>i</sub> is the ''i''-th prime number, and

:<math>a_1\geq a_2\geq\dotsb\geq a_k\geq 1.</math>

That is, they proved that if ''n'' is superabundant, the prime decomposition of ''n'' has non-increasing exponents (the exponent of a larger prime is never more than that a smaller prime) and that all primes up to <math>p_k</math> are factors of ''n''.  Then in particular any superabundant number is an even integer, and it is a multiple of the ''k''-th [[Primorial|primorial]] <math>p_k\#.</math>

In fact, the last exponent ''a''<sub>''k''</sub> is equal to 1 except when n is 4 or 36.

Superabundant numbers are closely related to [[Highly composite number|highly composite number]]s. Not all superabundant numbers are highly composite numbers. In fact, only 449 superabundant and highly composite numbers are the same {{OEIS|id=A166981}}. For instance, 7560 is highly composite but not superabundant. Conversely, 1163962800 is superabundant but not highly composite.

Alaoglu and Erdős observed that all superabundant numbers are [[Highly abundant number|highly abundant]]. 

Not all superabundant numbers are [[Harshad number]]s. The first exception is the 105th superabundant number, 149602080797769600.  The digit sum is 81, but 81 does not divide evenly into this superabundant number.

Superabundant numbers are also of interest in connection with the [[Riemann hypothesis]], and with Robin's theorem that the Riemann hypothesis is equivalent to the statement that
:<math>\frac{\sigma(n)}{e^\gamma n\log\log n} < 1</math>
for all ''n'' greater than the largest known exception, the superabundant number 5040. If this inequality has a larger counterexample, proving the Riemann hypothesis to be false, the smallest such counterexample must be a superabundant number {{harv|Akbary|Friggstad|2009}}.

Not all superabundant numbers are [[Colossally abundant number|colossally abundant]].

== Extension ==
The '''generalized <math>k</math>-super abundant numbers''' are those such that <math>\frac{\sigma_k(m)}{m^k} < \frac{\sigma_k(n)}{n^k}</math> for all <math>m < n</math>, where <math>\sigma_k(n)</math> is the sum of the <math>k</math>-th powers of the divisors of <math>n</math>.

1-super abundant numbers are superabundant numbers. 0-super abundant numbers are highly composite numbers.

For example, generalized 2-super abundant numbers are 1, 2, 4, 6, 12, 24, 48, 60, 120, 240, ... {{OEIS|id=A208767}}

== References ==
*{{citation|first1=Keith|last1=Briggs|title=Abundant numbers and the Riemann hypothesis|journal=Experimental Mathematics|volume=15|year=2006|issue=2 |pages=251–256|doi=10.1080/10586458.2006.10128957 |s2cid=46047029 |url=http://projecteuclid.org/euclid.em/1175789744 }}.
*{{citation|doi=10.4169/193009709X470128|first1=Amir|last1=Akbary|first2=Zachary|last2=Friggstad|title=Superabundant numbers and the Riemann hypothesis|journal=American Mathematical Monthly|volume=116|issue=3|year=2009|pages=273–275}}.
*{{citation|first1=Leonidas|last1=Alaoglu|first2=Paul|last2=Erdős|year=1944|title=On highly composite and similar numbers|journal=[[Organization:Transactions of the American Mathematical Society|Transactions of the American Mathematical Society]]|volume=56|issue=3|pages=448–469|doi=10.2307/1990319|publisher=American Mathematical Society|jstor=1990319}}.

== External links ==
* [http://mathworld.wolfram.com/SuperabundantNumber.html MathWorld: Superabundant number]


{{Divisor classes}}
{{Classes of natural numbers}}

[[Category:Divisor function]]
[[Category:Integer sequences]]

{{Sourceattribution|Superabundant number}}