IsPerfect

Tests whether $n$ equals the sum of its proper divisors.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPerfect number↗MathWorldPerfectNumber↗WikidataQ170043↗Britannicatopic/perfect-number↗OEISA000396↗Wolfram LanguagePerfectNumberQ!↗

IsPerfect(n)$n = \sigma_1(n) - n$

Typed signatures

Type	Package	Applies
(integer) -> boolean	compute-engine	always
(integer) -> boolean	number-theoryreplaces compute-engine	always

Domain: Number theory

Details

• compute-engine has this natively; the gap closed here is negative $n$ (never perfect) and threading over a list.
• The even perfect numbers are exactly $2^{p-1}(2^p-1)$ for a Mersenne prime exponent $p$ — see PerfectNumber and MersennePrimeExponent. Whether an odd perfect number exists is open.
• Wolfram's PerfectNumberQ.

Examples

Basic #

The smallest perfect number: $1+2+3=6$

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In$\mathrm{IsPerfect}(6)$Out$\mathrm{True}$

Abundant, not perfect: its proper divisors sum to 16

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In$\mathrm{IsPerfect}(12)$Out$\mathrm{False}$

Scope #

Listable

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In$\mathrm{IsPerfect}(\bigl\lbrack6, 28, 12\bigr\rbrack)$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{False}\bigr\rbrack$

Applications #

The third perfect number, $2^4(2^5-1)$

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In$\mathrm{IsPerfect}(496)$Out$\mathrm{True}$

The second perfect number, $1+2+4+7+14$

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In$\mathrm{IsPerfect}(28)$Out$\mathrm{True}$

Possible issues #

Negative numbers are not perfect

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In$\mathrm{IsPerfect}(-6)$Out$\mathrm{False}$

An ordinary large number is not perfect

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In$\mathrm{IsPerfect}(1\,234\,121)$Out$\mathrm{False}$

Stays symbolic for a non-numeric argument

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In$\mathrm{IsPerfect}(\mathrm{Simplify}(x))$Out$\mathrm{IsPerfect}(x)$

About

Examples	8
Checked against	Wolfram Language 6 agree2 differ
Crosswalk	WikipediaMathWorldWikidataBritannicaOEISWolfram Language

See also: PerfectNumber, MersennePrimeExponent, DivisorSigma