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.