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# abundant number

An integer $n$ is an abundant number if the sum of the proper divisors of $n$ is more than $n$ itself, or the sum of all the divisors is more than twice $n$. That is, $\sigma(n)>2n$, with $\sigma(n)$ being the sum of divisors function.

For example, the integer 30. Its proper divisors are 1, 2, 3, 5, 6, 10, 15, which add up to 42.

Multiplying a perfect number by some integer $x$ gives an abundant number (as long as $x>1$).

Given a pair of amicable numbers, the lesser of the two is abundant, its proper divisors adding up to the greater of the two.

Related:

AmicableNumbers

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## Mathematics Subject Classification

11A05*no label found*

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## Info

## Attached Articles

theorem on multiples of abundant numbers by CompositeFan

abundance by CompositeFan

quasiperfect number by CompositeFan

every even integer greater than 46 is the sum of two abundant numbers by PrimeFan

positive multiple of an abundant number is abundant by Mathprof

every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers by rspuzio

formula for sum of divisors by rspuzio

colossally abundant number by CompositeFan

every even integer greater than 70 is the sum of two abundant numbers in more than one way by PrimeFan

abundance by CompositeFan

quasiperfect number by CompositeFan

every even integer greater than 46 is the sum of two abundant numbers by PrimeFan

positive multiple of an abundant number is abundant by Mathprof

every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers by rspuzio

formula for sum of divisors by rspuzio

colossally abundant number by CompositeFan

every even integer greater than 70 is the sum of two abundant numbers in more than one way by PrimeFan

## Corrections

## Versions

(v6) by CompositeFan 2013-03-22