A note on primes with prime indices

TL;DR

This paper investigates the properties of recursively defined prime sequences with prime indices, providing bounds, answering open questions, and analyzing their counting functions to deepen understanding of their behavior.

Contribution

It offers explicit bounds and answers to questions about prime sequences with prime indices, advancing knowledge on their structure and growth.

Findings

Established explicit upper and lower bounds for $p_{n}^{(k)}$

Answered questions and proved a conjecture by Miska and Tóth

Analyzed the counting functions of the recursive prime sequences

Abstract

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