Definition:Abundant Number

Definition

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $\map A n$ denote the abundance of $n$.

Definition 1

Let $A \left({n}\right)$ denote the abundance of $n$.

$n$ is abundant if and only if $A \left({n}\right) > 0$.

Definition 2

Let $\sigma \left({n}\right)$ be the sigma function of $n$.

$n$ is abundant if and only if $\dfrac {\sigma \left({n}\right)} n > 2$.

Definition 3

$n$ is abundant if and only if it is smaller than its aliquot sum.

Sequence

The sequence of abundant numbers begins:

$12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, \ldots$

Also see

• Results about abundant numbers can be found here.