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# Superabundant number

In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. Formally, a natural number n is called superabundant iff for any m < n,

$\frac{\sigma(m)}{m} < \frac{\sigma(n)}{n}$

where σ denotes the divisor 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, ... ; superabundant numbers are closely related to highly composite numbers.

Superabundant numbers were first defined in [AlaErd44].

 Contents

## Properties

Alaoglu and Erdős proved [AlaErd44] that if n is superabundant, then there exist a2, ..., ap such that

$n=\prod_{i=2}^pi^{a_i}$

and

$a_2\geq a_3\geq\dots\geq a_p$

In fact, ap is nearly always 1.

It can also be shown that all superabundant numbers are Harshad numbers.