On perfect and near-perfect numbers

Paul Pollack, Vladimir Shevelev

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


We call n a near-perfect number if n is the sum of all of its proper divisors, except for one of them, which we term the redundant divisor. For example, the representation12=1+2+3+6 shows that 12 is near-perfect with redundant divisor 4. Near-perfect numbers are thus a very special class of pseudoperfect numbers, as defined by Sierpiński. We discuss some rules for generating near-perfect numbers similar to Euclid's rule for constructing even perfect numbers, and we obtain an upper bound of x 5/6+o(1) for the number of near-perfect numbers in [1, x], as x→∞.

Original languageEnglish
Pages (from-to)3037-3046
Number of pages10
JournalJournal of Number Theory
Issue number12
StatePublished - 1 Dec 2012


  • Perfect number
  • Pseudoperfect number
  • Sum-of-divisors function

ASJC Scopus subject areas

  • Algebra and Number Theory


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