I was reading about the reason why the reciprocals of the primes have a divergent sum. So I was thinking of changing the index to the $k$th prime. We get: $$\sum_{k=1}^\infty \frac{1}{p_{p_k}}=S$$ When I first posted this here, I was thinking that it was a divergent series like its predecessor. But @GregMartin pointed out that this sum actually converges. So I was wondering if this number is irrational? Could it also be transcendental? I am afraid this question is as hard as proving that Euler's constant $\gamma$ is transcendental.
Edit: I understand that this question is very hard to solve. So maybe is there any explanation why it might not be solvable by current methods?