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# Abundant number

## Definition

An abundant number is any number, n, which has a sigma value greater than 2n.

## Example

The divisors of 12 are $\displaystyle{ (1, 2, 3, 4, 6, 12) }$, so

$\displaystyle{ \sigma(12)\ =\ 1+2+3+4+6+12\ =\ 28 }$

## Abundant numbers and aliquot sequences

Abundant numbers increase the size of an aliquot sequence because when an abundant number occurs in a sequence, the next step is larger than the current step. Also, when a sequence is controlled by a driver, the subsequent steps are always abundant until an escape from the driver is obtained.