A note on primes with prime indices

Let $n,k\in \mathbb{N}$ and let $p_n$ denote the $n$th prime number. We define $p_n^{(k)}$ recursively as $p_n^{(1)}:=p_n$ and $p_n^{(k)}=p_{p_n^{(k-1)}}$, that is, $p_n^{(k)}$ is the $p_n^{(k-1)}$th prime. In this note we give answers to some questions and prove a conjecture posed by Miska and T'{o}th in their recent paper concerning subsequences of the sequence of prime numbers. In particular, we establish explicit upper and lower bounds for $p_n^{(k)}$. We also study the behaviour of the counting functions of the sequences $(p_n^{(k)}{)}_{k=1}^{\mathrm{\infty }}$ and $(p_k^{(k)}{)}_{k=1}^{\mathrm{\infty }}$.