hyperperfect number

A hyperperfect number n for a given k is an integer such that kσ(n)=-1+k+(k+1)n, where σ(x) is the sum of divisors function. n is then called k-hyperperfect. For example, 325 is 3-hyperperfect since its divisorsMathworldPlanetmathPlanetmath (1, 5, 13, 25, 65, 325) add up to 434, and 3×434=-1+3+(3+1)325=1302. Numbers that are 1-hyperperfect are by default called perfect numbers, since 1σ(n)=-1+1+(1+1)n=2n.

The 2-hyperperfect numbers are listed in A007593 of Sloane’s OEIS. As of 2007, the only known 3-hyperperfect number is 325. The two known 4-hyperperfect numbers are 1950625 and 1220640625, a sequenceMathworldPlanetmath too short to list in the OEIS, and no 5-hyperperfect numbers are known to exist. The 6-hyperperfect numbers are listed in A028499.


  • 1 Judson S. McCrainie, “A Study of Hyperperfect Numbers” Journal of Integer Sequences 3 (2000): 00.1.3
Title hyperperfect number
Canonical name HyperperfectNumber
Date of creation 2013-03-22 17:49:38
Last modified on 2013-03-22 17:49:38
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 4
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A05