A note on primes with prime indices

2019/09/26 by Błażej Żmija, Żmija, Błażej
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1909.12139

openalex publication_date 2019/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let n,k∈ℕ and let p_n denote the nth 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^∞ and (p_k^(k))_k=1^∞.

Citations

• Approximate formulas for some functions of prime numbers

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