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# colossally abundant number

An integer $n$ is a colossally abundant number if there is an exponent $\epsilon>1$ 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,

$\frac{\sigma(n)}{n^{\epsilon}}\geq\frac{\sigma(k)}{k^{\epsilon}},$ |

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 $i=\Omega(c_{i})$ is true.

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(v4) by CompositeFan 2013-03-22