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 < 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 were first defined by Leonidas Alaoglu and Paul Erdős (1944).

[edit] Properties

Leonidas Alaoglu and Paul Erdős (1944) proved that if n is superabundant, then there exist an i and a₁, a₂, a₃..., a_i such that

$n=\prod_{l=1}^i (p_l)^{a_l}$

where p_l is the l-th prime number, and

$a_1\geq a_2\geq\dots\geq a_i$

Or to wit, if n is superabundant then the prime decomposition of n has decreasing exponents, the smaller the prime, the larger its exponent.

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

Superabundant numbers are closely related to highly composite numbers. All superabundant numbers are highly composite numbers, but not every highly composite number is superabundant. For instance, 7560 is highly composite but not superabundant. Alaoglu and Erdős observed that all superabundant numbers are highly abundant. It can also be shown that all superabundant numbers are Harshad numbers.

Superabundant numbers are also of interest in connection with the Riemann hypothesis, and with Robin's theorem that the Riemann hypothesis is equivalent to the statement that

$\frac{\sigma(n)}{e^\gamma n\log\log n} < 1$

for all n greater than the largest known exception, the superabundant number 5040. If this inequality has a larger counterexample, proving the Riemann hypothesis to be false, the smallest such counterexample must be a superabundant number (Akbary & Friggstad 2009).

[edit] References

Akbary, Amir; Friggstad, Zachary (2009), "Superabundant numbers and the Riemann hypothesis", American Mathematical Monthly 116 (3): 273–275 .
Alaoglu, Leonidas; Erdős, Paul (1944), "On highly composite and similar numbers", Transactions of the American Mathematical Society 56 (3): 448–469, doi:10.2307/1990319 .