Equivalence of Definitions of Abundant Number

Theorem
The following definitions of a abundant number are equivalent:

Proof
By definition of abundance:


 * $A \left({n}\right) = \sigma \left({n}\right) - 2 n$

By definition of $\sigma$ function:
 * $\sigma \left({n}\right)$ is the sum of all the divisors of $n$.

Thus $\sigma \left({n}\right) - n$ is the sum of the proper divisors of $n$.

The result follows.