Superabundant number

A natural number n is called superabundant precisely when, for all m < n: where σ denotes the sum-of-divisors function (i.e., the sum of all positive divisors of n, including n itself).

Superabundant numbers were defined by Leonidas Alaoglu and Paul Erdős (1944).

Unknown to Alaoglu and Erdős, about 30 pages of Ramanujan's 1915 paper "Highly Composite Numbers" were suppressed.

Those pages were finally published in The Ramanujan Journal 1 (1997), 119–153.

are factors of n. Then in particular any superabundant number is an even integer, and it is a multiple of the k-th primorial

In fact, the last exponent ak is equal to 1 except when n is 4 or 36.

In fact, only 449 superabundant and highly composite numbers are the same (sequence A166981 in the OEIS).

Alaoglu and Erdős observed that all superabundant numbers are highly abundant.

The digit sum is 81, but 81 does not divide evenly into this superabundant number.

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 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).

For example, generalized 2-super abundant numbers are 1, 2, 4, 6, 12, 24, 48, 60, 120, 240, ... (sequence A208767 in the OEIS)