Abundant number

An abundant number is a natural number n for which the sum of divisors σ(n) satisfies σ(n) > 2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) satisfies s(n) > n. The abundance of a natural number is the integer σ(n) − 2n (equivalently, s(n) − n).

The first 28 abundant numbers are: For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36.

The first known classification of numbers as deficient, perfect or abundant was by Nicomachus in his Introductio Arithmetica (circa 100 AD), which described abundant numbers as like deformed animals with too many limbs.

The sequence (ak) of least numbers n such that σ(n) > kn, in which a2 = 12 corresponds to the first abundant number, grows very quickly (sequence A134716 in the OEIS).

The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 33 × 52 × 72 × 11 × 13 × 17 × 19 × 23 × 29.