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 Tóth 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 $(p_{n}^{(k)})_{k=1}^{\infty}$ and $(p_{k}^{(k)})_{k=1}^{\infty}$.

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