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On the number of special numbers

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Abstract

For lack of a better word, a number is called special if it has mutually distinct exponents in its canonical prime factorizaton for all exponents. Let V(x) be the number of special numbers ≤x. We will prove that there is a constant c>1 such that \(\displaystyle {V(x) \sim \frac {cx}{\log x}}\). We will make some remarks on determining the error term at the end. Using the explicit abc conjecture, we will study the existence of 23 consecutive special integers.

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Acknowledgements

We thank Shanta Laishram, T N Shorey and the referee for helpful comments. The research of the second author was partially supported by an NSERC Discovery grant.

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Authors and Affiliations

  1. Department of Mathematics Education, Gazi University, Teknikokullar, Ankara, 06500, Turkey

    KEVSER AKTAŞ

  2. Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada

    M RAM MURTY

Authors
  1. KEVSER AKTAŞ
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  2. M RAM MURTY
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Correspondence to M RAM MURTY.

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Communicating Editor: B Sury

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AKTAŞ, K., MURTY, M.R. On the number of special numbers. Proc Math Sci 127, 423–430 (2017). https://doi.org/10.1007/s12044-016-0326-z

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  • DOI: https://doi.org/10.1007/s12044-016-0326-z

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