Definition:Abundance

Let $$n$$ 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 divisor of $$n$$.

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

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

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

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

(No such numbers are known.)

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

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

Also See
Compare 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.