# PerfectNumber

gives the n perfect number.

# Details • A perfect number is a positive integer that is equal to half the sum of its divisors.
• In , n must be a positive integer.
• As of this version of the Wolfram Language, only 51 perfect numbers are known. will attempt to find perfect numbers for any n, but cannot be expected to return results in a reasonable time for .
• PerfectNumber[n,"Even"] gives the n even perfect number. As of this version of the Wolfram Language, the first 47 even perfect numbers are known, and 4 more whose position n is not yet certain. PerfectNumber[n,"Even"] will attempt to find even perfect numbers for , but cannot be expected to return results in a reasonable time.
• PerfectNumber[n,"Odd"] will attempt to find the n odd perfect number. As of this version of the Wolfram Language, no odd perfect number is known, and PerfectNumber[n,"Odd"] cannot be expected to return a result. There are no odd perfect numbers less than the 18 even perfect number.

# Examples

open allclose all

## Basic Examples(1)

Return the first 10 perfect numbers:

## Properties & Relations(4)

Even perfect numbers are related to Mersenne prime exponents:

Even perfect numbers are triangular numbers related to Mersenne prime exponents:

Even perfect numbers are also hexagonal numbers related to Mersenne prime exponents:

All even perfect numbers greater than 6 are of the following form for some value of k:

Even perfect numbers end in either 6 or 28:

Plot the integer length of the first 47 even perfect numbers:

## Possible Issues(2)

As of this version of the Wolfram Language, no odd perfect number is known:

As of this version of the Wolfram Language, the first 47 even perfect numbers are known:

But in total, 51 even perfect numbers are known: