A+ CATEGORY SCIENTIFIC UNIT

On near-perfect and deficient-perfect numbers

Volume 133 / 2013

Min Tang, Xiao-Zhi Ren, Meng Li Colloquium Mathematicum 133 (2013), 221-226 MSC: Primary 11A25. DOI: 10.4064/cm133-2-8

Abstract

For a positive integer $n$, let $\sigma (n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a near-perfect number if $\sigma (n) = 2n + d$, and a deficient-perfect number if $\sigma (n) = 2n - d$. We show that there is no odd near-perfect number with three distinct prime divisors and determine all deficient-perfect numbers with at most two distinct prime factors.

Authors

  • Min TangDepartment of Mathematics
    Anhui Normal University
    Wuhu 241003, China
    e-mail
  • Xiao-Zhi RenSchool of Mathematical Sciences
    Nanjing Normal University
    Nanjing 210023, China
    e-mail
  • Meng LiDepartment of Mathematics
    Anhui Normal University
    Wuhu 241003, China
    e-mail

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