superabundant number

In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. A natural number ''n'' is called superabundant precisely when, for all ''m'' < ''n''

:$\frac < \frac$

where ''σ'' denotes the sum-of-divisors function (i.e., the sum of all positive divisors of ''n'', including ''n'' itself). The first few superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... . For example, the number 5 is not a superabundant number because for 1, 2, 3, 4, and 5, the sigma is 1, 3, 4, 7, 6, and 7/4 > 6/5.

Superabundant numbers were defined by . 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 numbers, which include the superabundant numbers.

Properties

proved that if ''n'' is superabundant, then there exist a ''k'' and ''a''₁, ''a''₂, ..., ''a''_''k'' such that

:$n=\prod_^k (p_i)^$

where ''p''_i is the ''i''-th prime number, and

:$a_1\geq a_2\geq\dotsb\geq a_k\geq 1.$

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 $p_k$ are factors of ''n''. Then in particular any superabundant number is an even integer, and it is a multiple of the ''k''-th primorial $p_k\#.$

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

Superabundant numbers are closely related to