# Superabundant number

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 precisely when, for any "m" &lt; "n",

:$frac\left\{sigma\left(m\right)\right\}\left\{m\right\} < frac\left\{sigma\left(n\right)\right\}\left\{n\right\}$

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, ... OEIS|id=A004394; superabundant numbers are closely related to highly composite numbers. All superabundant numbers are highly composite numbers, but 7560 is a counterexample of the converse.

Superabundant numbers were first defined in [AlaErd44] .

Properties

Leonidas Alaoglu and Paul Erdős proved [AlaErd44] that if "n" is superabundant, then there exist "a"2, ..., "a""p" such that

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

and

:$a_2geq a_3geqdotsgeq a_p$

In fact, "a""p" is equal to 1 except when n is 4 or 36.

Alaoglu and Erdős observed that all superabundant numbers are highly abundant. It can also be shown that all superabundant numbers are Harshad numbers.

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

References

* [AlaErd44] - Leonidas Alaoglu and Paul Erds, "On Highly Composite and Similar Numbers", Trans. AMS 56, 448-469 (1944)

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