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:
- $\map {\sigma_1} n= 2 n$
where $\sigma_1: \Z_{>0} \to \Z_{>0}$ is the divisor sum 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.
They were actively investigated between $1631$ and $1647$ by Mersenne, Pierre de Fermat, André Jumeau, Bernard Frénicle de Bessy and Descartes.
Many new ones have been found since.
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
- 1919: Leonard Eugene Dickson: History of the Theory of Numbers: Volume $\text { I }$ ... (previous) ... (next): Preface
- 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. https://mathworld.wolfram.com/MultiperfectNumber.html