Equivalence of Definitions of Abundant Number

Theorem

The following definitions of a abundant number are equivalent:

Definition 1

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

$n$ is abundant if and only if $\map A n > 0$.

Definition 2

Let $\map {\sigma_1} n$ be the divisor sum function of $n$.

$n$ is abundant if and only if $\dfrac {\map {\sigma_1} n} n > 2$.

Definition 3

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

Proof

By definition of abundance:

$\map A n = \map {\sigma_1} n - 2 n$

By definition of divisor sum function:

$\map {\sigma_1} n$ is the sum of all the divisors of $n$.

Thus $\map {\sigma_1} n - n$ is the aliquot sum of $n$.

The result follows.

$\blacksquare$