Power of 2 is Almost Perfect

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Theorem

Let $n \in \Z_{>0}$ be a power of $2$:

$n = 2^k$

for some $k \in \Z_{>0}$.


Then $n$ is almost perfect.


Proof

\(\ds \map A n\) \(=\) \(\ds \dfrac {2^k} {2 - 1} - 2^k - \dfrac 1 {2 - 1}\) Power of Prime is Deficient
\(\ds \) \(=\) \(\ds 2^k - 2^k - 1\)
\(\ds \) \(=\) \(\ds - 1\)


and so $n = p^k$ is almost perfect by definition.

$\blacksquare$


Sources

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