An integer is an abundant number if the sum of the proper divisors of is more than itself, or the sum of all the divisors is more than twice . That is, , with being the sum of divisors function.

For example, the integer 30. Its proper divisors are 1, 2, 3, 5, 6, 10, 15, which add up to 42.

Multiplying a perfect number by some integer gives an abundant number (as long as ).

Given a pair of amicable numbers, the lesser of the two is abundant, its proper divisors adding up to the greater of the two.