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,

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 ^*.

Properties

Leonidas Alaoglu and Paul Erdős proved ^* that if n is superabundant, then there exist a₂, ..., a_p such that

$n=\backslash prod\_\{i=2\}^pi^\{a\_i\}$

and

$a\_2\backslash geq\; a\_3\backslash geq\backslash dots\backslash geq\; a\_p$

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

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

Integer sequences