On distribution of subsequences of primes having prime indices with respect to the $(R)$-denseness and convergence exponent

Abstract

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define subsequences $(p^{(k)}_n)_{n=1}^{+\infty}$ of the sequence $(p_n)_{n=1}^{+\infty}$ in the following way: let $p_n^{(1)}=p_n$ and $p_n^{(k+1)}=p_{p_n^{(k)}}$. In this paper we study and describe some interesting properties of the sets $\mathbb{P}_k=\{p_1^{(k)}<p_2^{(k)}<\dots<p_n^{(k)}<\dots\}$, $\mathbb{P}_n^{\mathrm{T}}=\{p_n^{(1)}<p_n^{(2)}<\dots<p_n^{(k)}<\dots\}$ and $\text{Diag}\mathbb{P}=\{p^{(1)}_1<p^{(2)}_2<\dots <p^{(k)}_k<\dots\}$ and their elements, for $k,n\in\mathbb{N}$. Especially, we check whether these sets have dense sets of ratios in $\mathbb{R}_+$. Moreover, we compute their exponents of convergence and asymptotics of their counting functions.