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 \(\log 2\).
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The authors would especially like to thank the anonymous reviewers for their valuable remarks and suggestions.
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Attila Kovács was supported by the Project no. TKP2020-NKA-06 (Application domain specific highly reliable IT solutions) with the support from the National Research, Development and Innovation Fund of Hungary, financed under the Thematic Excellence Programme funding scheme.
Norbert Tihanyi was supported by the Project no. TKP2021-NVA-29 with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.
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Borsos, B., Kovács, A. & Tihanyi, N. On reciprocal sums of infinitely many arithmetic progressions with increasing prime power moduli. Acta Math. Hungar. 171, 203–220 (2023). https://doi.org/10.1007/s10474-023-01385-9
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DOI: https://doi.org/10.1007/s10474-023-01385-9