Definition:Abundance

Definition

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 (Introduction to Arithmetic).