PerfectNumber

The $n$th perfect number.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguagePerfectNumber✓↗

PerfectNumber(n)$2^{p-1}(2^p-1)$ for the $n$th known Mersenne prime exponent $p$

Typed signatures

Type	Package	Applies
(integer) -> integer	number-theory	always

Domain: Number theory

Details

• Every even perfect number has this form (Euclid–Euler); whether an odd one exists is open. From the same table as MersennePrimeExponent, so it stays unevaluated past it.

Examples

Basic #

live ↗

In$\mathrm{PerfectNumber}(1)$Out$6$

live ↗

In$\mathrm{PerfectNumber}(3)$Out$496$

live ↗

In$\mathrm{PerfectNumber}(5)$Out$33\,550\,336$

Scope #

Listable

live ↗

In$\mathrm{PerfectNumber}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\bigl\lbrack6, 28, 496, 8\,128\bigr\rbrack$

The first ten perfect numbers as exact integers, from $6$ to a 54-digit value

live ↗

In$\mathrm{PerfectNumber}(1..10)$Out$\bigl\lbrack6, 28, 496, 8\,128, 33\,550\,336, 8\,589\,869\,056, 137\,438\,691\,328, 2\,305\,843\,008\,139\,952\,128, 2\,658\,455\,991\,569\,831\,744\,654\,692\,615\,953\,842\,176, 191\,561\,942\,608\,236\,107\,294\,793\,378\,084\,303\,638\,130\,997\,321\,548\,169\,216\bigr\rbrack$

About

Examples	5
Checked against	Wolfram Language 5 agree
Crosswalk	Wolfram Language

See also: MersennePrimeExponent, IsPerfect