---
tag: 0VS0MP1P5H
kind: definition
name: Abundant number
msc: [11A25]
defines: 0VG7VNEWMG
uses: [009QCP5T3Z]
sources:
  - url: https://en.wikipedia.org/wiki/Abundant_number
    locator: Definition and examples
---

## Definition

A positive integer $n$ is abundant if the sum of its positive divisors other than $n$ is greater than $n$.
Equivalently, $\sigma(n) > 2n$, where $\sigma$ is the [[sum of divisors|009QCP5T3Z]].

## Remarks

The smallest abundant number is $12$, since $1 + 2 + 3 + 4 + 6 = 16 > 12$.
The smallest odd abundant number is $945$.
Each multiple $kn$ with $k \ge 2$ of a [[perfect number|0SB7Q8SYSF]] $n$ is abundant.

## Explanations

### L1 human

An abundant number has "too many" divisors: its smaller divisors add up to more than the number.
Compare with [[perfect numbers|0SB7Q8SYSF]], where the sum is equal, and [[deficient numbers|0G01TS1RRP]], where it is smaller.