Challenge

Let

$A=\{a\in\mathbb N : a=n^m \text{ for some integers } n\geq 2,\ m\geq 2\}.$

Thus

$A=\{4,8,9,16,25,27,32,36,49,64,81,\ldots\}.$

Note that $A$ is a set, so every perfect power is counted only once. For example,

$64=2^6=4^3=8^2,$

but $64$ occurs only once in the sum.

Evaluate

.

Can you find an elegant proof?