Reciprocals of Prime-Primes

Calculus Level 2

Let $P_k$ be the $k^\text{th}$ prime: $P_1 = 2,$ $P_2 = 3,$ and so on. As it turns out, $\displaystyle \sum_{k=1}^{\infty} \frac{1}{P_k}=\frac{1}{2}+\frac{1}{3}+\frac{1}{5} + \frac{1}{7} + \cdots$ diverges.

Does $\displaystyle \sum_{k=1}^{\infty} \frac{1}{P_{P_k}} = \frac{1}{3}+\frac{1}{5}+\frac{1}{11}+\frac{1}{17} + \cdots$ converge?