colossally abundant number

An integer $n$ is a colossally abundant number if there is an exponent $ϵ>1$ such that the sum of divisors 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,

$$\frac{\sigma (n)}{n^{ϵ}}\ge \frac{\sigma (k)}{k^{ϵ}},$$

with $\sigma (n)$ 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 factors, that is to say that for the $i$th colossally abundant number $c_i$ the equality $i=\mathrm{\Omega }(c_i)$ 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