# Definition:Perfect Number/Definition 3

## Definition

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $A \left({n}\right)$ denote the abundance of $n$.

$n$ is perfect if and only if $A \left({n}\right) = 0$.

## Sequence

The sequence of perfect numbers begins:

 $\displaystyle 6$ $=$ $\displaystyle 2^{2 - 1} \times 2^2 - 1$ $\displaystyle 28$ $=$ $\displaystyle 2^{3 - 1} \times 2^3 - 1$ $\displaystyle 496$ $=$ $\displaystyle 2^{5 - 1} \times 2^5 - 1$ $\displaystyle 8128$ $=$ $\displaystyle 2^{7 - 1} \times 2^7 - 1$ $\displaystyle 33 \, 550 \, 336$ $=$ $\displaystyle 2^{13 - 1} \times 2^{13} - 1$ $\displaystyle 8 \, 589 \, 869 \, 056$ $=$ $\displaystyle 2^{17 - 1} \times 2^{17} - 1$