# Definition:Abundance

## Definition

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

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

That is, let $\sigma \left({n}\right)$ be the sum of all positive divisors of $n$.

Then the abundance of $n$ is defined as $A \left({n}\right) = \sigma \left({n}\right) - 2 n$.

### Abundant Number

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

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

### Perfect Number

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

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

### Quasiperfect Number

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

$n$ is quasiperfect if and only if $A \left({n}\right) = 1$.

### Almost Perfect Number

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

$n$ is almost perfect if and only if $A \left({n}\right) = -1$.

### Deficient Number

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

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

## Historical Note

The concepts of abundant and deficient appear to have originated with the Neo-Pythagorean school, in particular Nicomachus of Gerasa, who wrote fancifully on the subject in his Ἀριθμητικὴ εἰσαγωγή (Introduction to Arithmetic).