Power of 2 is Almost Perfect

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.

$\ds \map A n$

$=$

$\ds \dfrac {2^k} {2 - 1} - 2^k - \dfrac 1 {2 - 1}$

$\ds $

$=$

$\ds 2^k - 2^k - 1$

$\ds $

$=$

$\ds - 1$

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

$\blacksquare$

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