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$S$-exponential numbers

Volume 175 / 2016

Vladimir Shevelev Acta Arithmetica 175 (2016), 385-395 MSC: Primary 11A51; Secondary 11B05. DOI: 10.4064/aa8395-5-2016 Published online: 30 September 2016

Abstract

We prove that, for every set $S$ of positive integers containing 1 (finite or infinite), the density $h(E(S))$ of the set $E(S)$ of numbers that have prime factorizations with exponents only from $S$ exists, and we give an explicit formula for it. Further, we study the set of such densities for all $S$ and prove that it is a perfect set with a countable set of gaps which are some left-sided neighborhoods of the densities corresponding to all finite $S$ except for $S=\{1\}.$

Authors

  • Vladimir ShevelevDepartment of Mathematics
    Ben-Gurion University of the Negev
    Beer-Sheva 84105, Israel
    e-mail

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