Superabundant number

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Divisibility-based
sets of integers
Form of factorization:
Prime number
Composite number
Powerful number
Square-free number
Achilles number
Constrained divisor sums:
Perfect number
Almost perfect number
Quasiperfect number
Multiply perfect number
Hyperperfect number
Superperfect number
Unitary perfect number
Semiperfect number
Primitive semiperfect number
Practical number
Numbers with many divisors:
Abundant number
Highly abundant number
Superabundant number
Colossally abundant number
Highly composite number
Superior highly composite number
Other:
Deficient number
Weird number
Amicable number
Friendly number
Sociable number
Solitary number
Sublime number
Harmonic divisor number
Frugal number
Equidigital number
Extravagant number
See also:
Divisor function
Divisor
Prime factor
Factorization
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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 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, ... (sequence A004394 in OEIS); 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].

[edit] Properties

Leonidas Alaoglu and Paul 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 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.

[edit] External links

[edit] References

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