# abundant number

An integer $n$ is an abundant number if the sum of the proper divisors of $n$ is more than $n$ itself, or the sum of all the divisors is more than twice $n$. That is, $\sigma(n)>2n$, with $\sigma(n)$ 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 $x$ gives an abundant number (as long as $x>1$).

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

Title abundant number AbundantNumber 2013-03-22 15:52:21 2013-03-22 15:52:21 CompositeFan (12809) CompositeFan (12809) 6 CompositeFan (12809) Definition msc 11A05 AmicableNumbers