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Colossally abundant number

In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of natural number. Formally, a number n is colossally abundant iff there is an ε > 0 such that for all k > 1,

$\frac{\sigma(n)}{n^{1+\varepsilon}}\geq\frac{\sigma(k)}{k^{1+\varepsilon}}$

where σ denotes the divisor function. The first few colossally abundant numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, ... ; all colossally abundant number are also superabundant numbers, but the converse is not generally true.

Properties

All colossally abundant numbers are Harshad numbers.

Relation to the Riemann hypothesis

Colossally abundant numbers are related to the Riemann hypothesis (someone should fill in details for this).

Also see

Categories: Number theory

Last updated: 05-29-2005 10:54:26

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03-10-2013 05:06:04
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