Definition:Triperfect Number

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Definition

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


Sequence of Triperfect Numbers

The sequence of triperfect numbers begins:

$120, \quad 672, \quad 523 \, 776, \quad 459 \, 818 \, 240, \quad 1 \, 476 \, 304 \, 896, \quad 51 \, 001 \, 180 \, 160$

These are all the triperfect numbers that are currently known.


Examples

$120$ is Triperfect

$120$ is triperfect:

$\map {\sigma_1} {120} = 360 = 3 \times 120$


$672$ is Triperfect

$672$ is triperfect:

$\map {\sigma_1} {672} = 2016 = 3 \times 672$


$523 \, 776$ is Triperfect

$523 \, 776$ is triperfect:

$\map {\sigma_1} {523 \, 776} = 1 \, 571 \, 328 = 3 \times 523 \, 776$


$459 \, 818 \, 240$ is Triperfect

$459 \, 818 \, 240$ is triperfect:

$\map {\sigma_1} {459 \, 818 \, 240} = 1 \, 379 \, 454 \, 720 = 3 \times 459 \, 818 \, 240$


$1 \, 476 \, 304 \, 896$ is Triperfect

$1 \, 476 \, 304 \, 896$ is triperfect:

$\map {\sigma_1} {1 \, 476 \, 304 \, 896} = 4 \, 428 \, 914 \, 688 = 3 \times 1 \, 476 \, 304 \, 896$


$51 \, 001 \, 180 \, 160$ is Triperfect

$51 \, 001 \, 180 \, 160$ is triperfect:

$\map {\sigma_1} {51 \, 001 \, 180 \, 160} = 153 \, 003 \, 540 \, 480 = 3 \times 51 \, 001 \, 180 \, 160$


Also known as

Some sources hyphenate: tri-perfect.


Also see


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 integers $n$ whose divisors added up to an integer multiple of $n$.


Sources

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