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 $A \left({n}\right) = \sigma \left({n}\right) - 2 n$.

Perfect
$n$ is perfect $A \left({n}\right) = 0$.

Also see

 * Definition:Abundancy

Historical Note
The concepts of abundant and deficient appear to have originated with the Neo-Pythagorean school, in particular, who wrote fancifully on the subject in his Ἀριθμητικὴ εἰσαγωγή.