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On the number of weakly prime-additive numbers

  • Y.-G. Chen
  • J.-H. FangEmail author
Article
  • 18 Downloads

Abstract

A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors \(p_{1},\ldots, p_{t}\) of n and positive integers \(\alpha_{1}, \ldots , \alpha_{t}\) such that \(n = p_{1}^{\alpha_{1}}+ \cdots + p_{t}^{\alpha_{t}}\). Erdős and Hegyvári [2] proved that, for any prime p, there exists a weakly prime-additive number which is divisible by p. Recently, Fang and Chen [3] proved that for any given positive integer m, there are infinitely many weakly prime-additive numbers which are divisible by m with t = 3 if and only if \(8 \nmid m\). In this paper, we prove that for any given positive integer m, the number of weakly prime-additive numbers which are divisible by m and less than x is larger than \({\rm exp}(c({\rm log log} x)^{2}/ {\rm log log log} x)\) for all sufficiently large x, where c is a positive absolute constant. The constant c depends on the result on the least prime number in an arithmetic progression.

Key words and phrases

prime-additive number Chinese remainder theorem Dirichlet’s theorem representation of integers 

Mathematics Subject Classification

11A07 11A41 

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Notes

Acknowledgement

We sincerely thank the referee for valuable suggestions.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.School of Mathematical Sciences and Institute of MathematicsNanjing Normal UniversityNanjingP. R. China
  2. 2.Department of MathematicsNanjing University of Information Science and TechnologyNanjingP. R. China

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