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

Numbers of the form k· p^n+1 with the restriction k < p^n are called generalized Proth numbers. For a fixed prime p we denote them by 𝒯_p . The underlying structure of 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 𝒯_p is presented. All formulae were implemented and verified by the PARI/GP computer algebra system. We show that the asymptotic density of ⋃_p∈𝒫𝒯_p is log 2 .