In number theory, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.

Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on the subject was done by Alaoglu and Erdős (1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 10⁴, and showed that the number of highly abundant numbers less than any N is at least proportional to log² N.

Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n,

where σ denotes the sum-of-divisors function. The first few highly abundant numbers are

For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ.

The only odd highly abundant numbers are 1 and 3.

Although the first eight factorials are highly abundant, not all factorials are highly abundant. For example,

but there is a smaller number with larger sum of divisors,