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PerfectNumber[n]

gives the n^(th) perfect number.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
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PerfectNumber

PerfectNumber[n]

gives the n^(th) perfect number.

Details

  • A perfect number is a positive integer that is equal to half the sum of its divisors.
  • In PerfectNumber[n], n must be a positive integer.
  • As of this version of the Wolfram Language, only 51 perfect numbers are known. PerfectNumber[n] 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^(th) even perfect number. As of this version of the Wolfram Language, the first 48 even perfect numbers are known, and 3 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^(th) 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^(th) even perfect number.

Examples

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Basic Examples  (1)

Return the first 10 perfect numbers:

Wolfram Language code: Table[PerfectNumber[n], {n, 10}]

Scope  (1)

PerfectNumber automatically threads over lists:

Wolfram Language code: PerfectNumber[Range[10]]

Properties & Relations  (4)

Even perfect numbers are related to Mersenne prime exponents:

Wolfram Language code: n = 5; mpe = MersennePrimeExponent[n]
Wolfram Language code: PerfectNumber[n, "Even"] == 2 ^ (mpe - 1)(2 ^ mpe - 1)

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

Wolfram Language code: PerfectNumber[n, "Even"] == PolygonalNumber[2 ^ mpe - 1]

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

Wolfram Language code: PerfectNumber[n, "Even"] == PolygonalNumber[6, 2 ^ (mpe - 1)]

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

Wolfram Language code: number[k_] := 1 + 9PolygonalNumber[8k + 2];
Wolfram Language code: PerfectNumber[2, "Even"] == number[0]
Wolfram Language code: PerfectNumber[3, "Even"] == number[1]
Wolfram Language code: PerfectNumber[4, "Even"] == number[5]
Wolfram Language code: PerfectNumber[5, "Even"] == number[341]
Wolfram Language code: PerfectNumber[6, "Even"] == number[5461]

Even perfect numbers end in either 6 or 28:

Wolfram Language code: PerfectNumber[{1, 3, 5, 6}, "Even"]
Wolfram Language code: PerfectNumber[{2, 4, 7, 8}, "Even"]

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

Wolfram Language code: ListPlot[IntegerLength[PerfectNumber[Range[48], "Even"]], ScalingFunctions -> "Log", Filling -> Axis]

Possible Issues  (2)

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

Wolfram Language code: TimeConstrained[PerfectNumber[1, "Odd"], 10]

As of this version of the Wolfram Language, the first 50 even perfect numbers have definite ranking:

Wolfram Language code: TimeConstrained[PerfectNumber[51, "Even"], 10]

More perfect numbers are known, but their ranking is still unknown:

Wolfram Language code: n = {82589933, 136279841}; numbers = 2 ^ (n - 1)(2 ^ n - 1); AllTrue[numbers, # > PerfectNumber[50, "Even"]&]
Wolfram Language code: AllTrue[numbers, PerfectNumberQ]
Wolfram Research (2016), PerfectNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/PerfectNumber.html (updated 2024).

Text

Wolfram Research (2016), PerfectNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/PerfectNumber.html (updated 2024).

CMS

Wolfram Language. 2016. "PerfectNumber." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/PerfectNumber.html.

APA

Wolfram Language. (2016). PerfectNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PerfectNumber.html

BibTeX

@misc{reference.wolfram_2026_perfectnumber, author="Wolfram Research", title="{PerfectNumber}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/PerfectNumber.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_perfectnumber, organization={Wolfram Research}, title={PerfectNumber}, year={2024}, url={https://reference.wolfram.com/language/ref/PerfectNumber.html}, note=[Accessed: 12-June-2026]}