Superabundant number

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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 < \frac]

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, ... (sequence in OEIS); superabundant numbers are closely related to highly composite numbers.

Superabundant numbers were first defined in [AlaErd44].

Properties

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

[n=\prod_^pi^]

and

[a_2\geq a_3\geq\dots\geq a_p]

In fact, a_p is equal to 1 except when n is 4 or 36.

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

See also

• Colossally abundant number
• Highly abundant number
• Abundant number
• Deficient number

External links

• [MathWorld: Superabundant number]

References

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

 

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