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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

Last updated: 05-29-2005 10:54:26
03-10-2013 05:06:04