# 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.

## 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.

$\blacksquare$