Definition:Perfect Number/Definition 3

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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:

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


Also see