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On near-perfect numbers

Volume 194 / 2020

Peter Cohen, Katherine Cordwell, Alyssa Epstein, Chung-Hang Kwan, Adam Lott, Steven J. Miller Acta Arithmetica 194 (2020), 341-366 MSC: Primary 11A25; Secondary 11N25, 11B83. DOI: 10.4064/aa180821-11-10 Published online: 27 March 2020


The study of perfect numbers (numbers which equal the sum of their proper divisors) goes back to antiquity, and is responsible for some of the oldest and most popular conjectures in number theory. We investigate a generalization introduced by Pollack and Shevelev: $k$-near-perfect numbers. These are examples of the well-known pseudoperfect numbers first defined by Sierpiński, and are numbers that equal the sum of all but at most $k$ of their proper divisors. We establish the asymptotic order of $k$-near-perfect numbers for all integers $k\ge 4$, as well as some properties of related quantities.


  • Peter CohenDepartment of Mathematics
    Massachusetts Institute of Technology
    Cambridge, MA 02139, U.S.A.
  • Katherine CordwellDepartment of Computer Science
    Carnegie Mellon University
    Pittsburgh, PA 15213, U.S.A.
  • Alyssa EpsteinStanford Law School
    Stanford, CA 94305, U.S.A.
  • Chung-Hang KwanDepartment of Mathematics
    Columbia University
    in the City of New York
    New York, NY 10027. U.S.A.
  • Adam LottDepartment of Mathematics
    University of California, Los Angeles
    Los Angeles, CA 90095, U.S.A.
  • Steven J. MillerDepartment of Mathematics and Statistics
    Williams College
    Williamstown, MA 01267, U.S.A.

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