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On consecutive integers divisible by the number of their divisors

Volume 173 / 2016

Titu Andreescu, Florian Luca, M. Tip Phaovibul Acta Arithmetica 173 (2016), 269-281 MSC: Primary 11N25; Secondary 11N41, 11N64. DOI: 10.4064/aa8242-1-2016 Published online: 14 April 2016


We prove that there are no strings of three consecutive integers each divisible by the number of its divisors, and we give an estimate for the number of positive integers $n\le x$ such that each of $n$ and $n+1$ is a multiple of the number of its divisors.


  • Titu AndreescuDepartment of Science/Mathematics Education
    The University of Texas at Dallas
    FN32, 800 West Campbell Road
    Richardson, TX 75080-3021, U.S.A.
  • Florian LucaSchool of Mathematics
    University of the Witwatersrand
    Private Bag X3, Wits 2050
    Johannesburg, South Africa
  • M. Tip PhaovibulP.O. Box 749
    Mosier, OR 97040, U.S.A.

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