SILK-ROAD 2003 Problem 4

Find $\sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \}$ .

Problem was post earlier here , but solution not gives and olympiad doesn't indicate, so I post it again :blush:

Official solution here