# Definition:Multiply Perfect Number

## 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 $n$.

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

## 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.