11A05-ColossallyAbundantNumber.tex

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\pmtitle{colossally abundant number}
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An integer $n$ is a {\em 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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