multiply perfect number

A multiply perfect number n for a given k is an integer such that σ(n)=kn, where σ(x) is the sum of divisors function. n is then called k-perfect. For example, 120 is 3-perfect since its divisorsMathworldPlanetmathPlanetmath (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120) add up to 360, which is thrice 120. Numbers that are 2-perfect are by default called perfect numbers.

The 3-perfect numbers are listed in A005820 of Sloane’s OEIS; A027687 lists 4-perfect numbers; etc. The first k-perfect number for k>1 are listed in A007539.

As of 2007, more than 5000 multiply perfect numbers were known. The Multiply Perfect Numbers webpage, hosted by Bielefeld University, provides an ASCII text file database giving the name of the discoverer, “abundancy” (the value of k for the particular n), some of the prime factorsMathworldPlanetmath and the value of loglogn.


  • 1 A. H. Beiler, Recreations in the Theory of Numbers. New York (1964): 22.
  • 2 A. L. Brown, “Multiperfect numbers” Scripta Math. 20 (1954): 103 - 106
Title multiply perfect number
Canonical name MultiplyPerfectNumber
Date of creation 2013-03-22 17:47:58
Last modified on 2013-03-22 17:47:58
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 5
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A05
Synonym multiperfect number
Synonym pluperfect number