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
A positive integer $n$ is classified as abundant iff $A \left({n}\right) > 0$.

Perfect
A positive integer $n$ is classified as perfect iff $A \left({n}\right) = 0$.

Quasiperfect
A positive integer $n$ is classified as quasiperfect iff $A \left({n}\right) = 1$.

(No such numbers are known.)

Almost Perfect
A positive integer $n$ is classified as almost perfect iff $A \left({n}\right) = -1$.

Deficient
A positive integer $n$ is classified as deficient iff $A \left({n}\right) < 0$.

Also see

 * Definition:Abundancy

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