Let $z(n)$ be the rank of apparition in the Fibonacci sequence, i.e., $z(n)$ is the smallest positive integer $k$ such that $n$ divides $F_k$. A well-known result is that $z(n)\leq 2n$, for all $n\geq 1$ with equality iff $n=6\cdot 5^k$, for $k\geq 0$.
I would like to prove that the series $$ \displaystyle\sum_{n\geq 1}\displaystyle\frac{z(n)}{n^2} $$ diverges.
Any help is welcome. Thanks!