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