Please help me evaluate the following infinite sum: $\sum\limits_{k=1}^{\infty}\frac{1}{p_k(p_k-1)}$ I'm trying to evaluate:
$\displaystyle{\sum_{k=1}^{\infty}\frac{1}{p_k(p_k-1)}}$,
where $p_k$ is the $k$-th prime number.
But I cannot even figure out how to begin. I have a feeling that this could involve the prime zeta function, but I'm not sure. In fact, I'm not even sure an analytic closed-form solution is possible. Could you please help?

We can rewrite this as $\sum \limits_{k=1}^\infty ( \frac{1}{p_k^2}+\frac{1}{p_k^3}+\dots)$.
So we want $\sum\limits_{k=2}^\infty P(k)$ where $P$ is the prime zeta function.
