colossally abundant number

An integer n is a colossally abundant number if there is an exponent ϵ>1 such that the sum of divisorsMathworldPlanetmathPlanetmath of n divided by n raised to that exponent is greater than or equal to the sum of divisors of any other integer k>1 divided by k raised to that same exponent. That is,


with σ(n) being the sum of divisors function.

The first few colossally abundant numbers are 1, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320. The index of a colossally abundant number is equal to the number of its nondistinct prime factorsMathworldPlanetmath, that is to say that for the ith colossally abundant number ci the equality i=Ω(ci) is true.

Title colossally abundant number
Canonical name ColossallyAbundantNumber
Date of creation 2013-03-22 17:37:52
Last modified on 2013-03-22 17:37:52
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 4
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A05