On reciprocal sums of infinitely many arithmetic progressions with increasing prime power moduli

Abstract

Numbers of the form \(k\cdot p^n+1\) with the restriction \(k < p^n\) are called generalized Proth numbers. For a fixed prime p we denote them by \(\mathcal{T}_p\). The underlying structure of \(\mathcal{T}_2\) (Proth numbers) was investigated in [2]. In this paper the authors extend their results to all primes. An efficiently computable upper bound for the reciprocal sum of primes in \(\mathcal{T}_p\) is presented. All formulae were implemented and verified by the PARI/GP computer algebra system. We show that the asymptotic density of \( \bigcup_{p\in \mathcal{P}} \mathcal{T}_p\) is \(\log 2\).