# Definition:Abundancy Index

## Definition

Let $n$ be a positive integer.

Let $\map \sigma n$ be the sigma function of $n$.

That is, let $\map \sigma n$ be the sum of all positive divisors of $n$.

Then the abundancy index of $n$ is defined as $\dfrac {\map \sigma n} n$.

### Abundant Number

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

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

### Perfect Number

A perfect number $n$ is a (strictly) positive integer such that:

$\dfrac {\sigma \left({n}\right)} n = 2$

where $\sigma: \Z_{>0} \to \Z_{>0}$ is the sigma function.

### Deficient Number

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

$n$ is deficient if and only if:

$\dfrac {\sigma \left({n}\right)} n < 2$