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
- Definition:Perfect Number
- Definition:Multiply Perfect Number
- Results about triperfect 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 integers $n$ whose divisors added up to an integer multiple of $n$.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $120$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $672$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $523,776$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $120$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $672$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $523,776$