- Superabundant number
In

mathematics , a**superabundant number**(sometimes abbreviated as**SA**) is a certain kind ofnatural number . Formally, a natural number "n" is called superabundant precisely when, for any "m" < "n",:$frac\{sigma(m)\}\{m\}\; <\; frac\{sigma(n)\}\{n\}$

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 number s. 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 andPaul Erdős proved [AlaErd44] that if "n" is superabundant, then there exist "a"_{2}, ..., "a"_{"p"}such that:$n=prod\_\{i=2\}^pi^\{a\_i\}$

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 number s.**External links*** [

*http://mathworld.wolfram.com/SuperabundantNumber.html MathWorld: Superabundant number*]**References*** [AlaErd44] -

Leonidas Alaoglu andPaul Erds , "On Highly Composite and Similar Numbers", Trans. AMS 56, 448-469 (1944 )

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