Abundant number

Definition

A positive integer $n$ is abundant if the sum of its positive divisors other than $n$ is greater than $n$. Equivalently, $\sigma(n) > 2n$, where $\sigma$ is the sum of divisors.

Remarks

The smallest abundant number is $12$, since $1 + 2 + 3 + 4 + 6 = 16 > 12$. The smallest odd abundant number is $945$. Each multiple $kn$ with $k \ge 2$ of a perfect number $n$ is abundant.

Explanations

L1 human

An abundant number has "too many" divisors: its smaller divisors add up to more than the number. Compare with perfect numbers, where the sum is equal, and deficient numbers, where it is smaller.