# Definition:Triperfect Number

## 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:

$\sigma \left({120}\right) = 360 = 3 \times 120$

### $672$ is Triperfect

$672$ is triperfect:

$\sigma \left({672}\right) = 2016 = 3 \times 672$

### $523 \, 776$ is Triperfect

$523 \, 776$ is triperfect:

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

### $459 \, 818 \, 240$ is Triperfect

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

$\sigma \left({459 \, 818 \, 240}\right) = 1 \, 379 \, 454 \, 720 = 3 \times 459 \, 818 \, 240$

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

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

$\sigma \left({1 \, 476 \, 304 \, 896}\right) = 4 \, 428 \, 914 \, 688 = 3 \times 1 \, 476 \, 304 \, 896$

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

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

$\sigma \left({51 \, 001 \, 180 \, 160}\right) = 153 \, 003 \, 540 \, 480 = 3 \times 51 \, 001 \, 180 \, 160$

## Also known as

Some sources hyphenate: tri-perfect.

## 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 $n$.