Definition:Abundancy Index

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 $$\frac {\sigma \left({n}\right)} n$$.

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

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

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

Also See
Compare Abundance.