Definition:Abundancy Index

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 divisors of $n$.

Then the abundancy of $n$ is defined as $\dfrac {\sigma \left({n}\right)} n$.

Abundant
A number is classified as abundant iff $\dfrac {\sigma \left({n}\right)} n > 2$.

Perfect
A number is classified as perfect iff $\dfrac {\sigma \left({n}\right)} n = 2$.

Deficient
A number is classified as deficient iff $\dfrac {\sigma \left({n}\right)} n < 2$.

Also See
Compare Abundance.