Convergence of a subseries of the series $\sum_{n \ge 1} \frac{1}{p_n}$, where $p_n$ is the $n$ th prime.

Question

Let $p_n$ be the $n$th prime number. Does the following series $$ \sum_{n \ge 1} \frac{1}{p_{p_n}} = \frac{1}{3} + \frac{1}{5} + \frac{1}{11} + \cdots $$ converge or diverge? Similarly, I am so curious about the convergence of the subsequent type of subserious, like $$ \sum_{n \ge 1} \frac{1}{p_{p_{p_n}}} $$, and so on.