# Definition:Multiply Perfect Number

## Contents

## Definition

A **multiply perfect number** is a positive integer $n$ such that the sum of its divisors is equal to an integer multiple of $n$.

## Also known as

Some sources hyphenate: **multiply-perfect**.

Other terms used:

**multiperfect****pluperfect**.

## Order of Multiply Perfect Number

Let $n \in \Z_{>0}$ be a **multiply perfect number** such that the sum of its divisors is equal to $m \times n$.

Then $n$ is **multiply perfect of order $m$**.

## Instances of Multiply Perfect Numbers

### Perfect Number

A **perfect number** $n$ is a (strictly) positive integer such that:

- $\sigma \left({n}\right) = 2 n$

where $\sigma: \Z_{>0} \to \Z_{>0}$ is the sigma function.

### Triperfect Number

A **triperfect number** is a positive integer $n$ such that the sum of its divisors is equal to $3$ times $n$.

### Quadruply Perfect Number

A **quadruply perfect number** is a positive integer $n$ such that the sum of its divisors is equal to $4$ times $n$.

## Also see

- Definition:Perfect Number
- Results about
**multiply perfect numbers**can be found here.

## Historical Note

Marin Mersenne was the first to discover the smallest **triperfect number** $120$.

He suggested to René Descartes that it would be an interesting exercise to hunt down further examples of **multiply perfect numbers**.

## Linguistic Note

Note that the word **multiply** in the term **multiply perfect number** is an adverb: a word that qualifies an adjective.

As such it should be interpreted as **multiple-ly**, that is, **in the form of being a multiple**, and is pronounced something like ** mul-ti-plee**.

Do not confuse with the verb form of **multiply**, meaning **to perform an act of multiplication**, which is pronounced something like ** mul-ti-pligh**.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $120$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $120$

- Weisstein, Eric W. "Multiperfect Number." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/MultiperfectNumber.html