superperfect number

A k-superperfect number n is an integer such that σk(n)=2n, where σk(x) is the iterated sum of divisors function. For example, 16 is 2-superperfect since its divisorsMathworldPlanetmathPlanetmath add up to 31, and in turn the divisors of 31 add up to 32, which is twice 16.

At first Suryanarayana only considered 2-superperfect numbers. It is easy to prove that numbers of the form 2p-1 are 2-superperfect only if 2p-1 is a Mersenne primeMathworldPlanetmath. The existence of odd 2-superperfect numbers appears as unlikely as that of regular odd perfect numbers.

Later, Dieter Bode generalized the concept for any k and proved that when k>2 there are no even k-superperfect numbers. Others have further generalized the concept to (k,m)-superperfect numbers satisifying the equality σk(n)=mn, and Weisstein programs a Mathematica command to default to m=2 when the third argument is omitted. For example, 8, 21, and 512 are (2, 3)-superperfect, since the second iteration of the sum of divisors function gives thrice them, 24, 63, and 1536 respectively.

Not to be confused with hyperperfect numbers, which satisfy a different equality involving the sum of divisors function.


  • 1 R. K. Guy, Unsolved Problems in Number TheoryMathworldPlanetmathPlanetmath New York: Springer-Verlag 2004: B9
  • 2 D. Suryanarayana, “Super perfect numbersElem. Math. 24 (1969): 16 - 17
  • 3 E. Weisstein, “ number” Mathworld
Title superperfect number
Canonical name SuperperfectNumber
Date of creation 2013-03-22 17:03:38
Last modified on 2013-03-22 17:03:38
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 5
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A25