Equivalence of Definitions of Abundant Number

Theorem
The following definitions of a abundant number are equivalent:

Proof
By definition of abundance:


 * $\map A n = \map {\sigma_1} n - 2 n$

By definition of divisor sum function:
 * $\map {\sigma_1} n$ is the sum of all the divisors of $n$.

Thus $\map {\sigma_1} n - n$ is the aliquot sum of $n$.

The result follows.