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# Descartes number

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A **Descartes number** (also called an "**odd spoof perfect number**") is an odd number that is a "spoof perfect number," i.e. that would be an odd perfect number if some of its composite factors were treated as if they were "spoof-prime factors."

n |

^{[1]}

m |

k |

σ (k) |

k |

Equivalently, we want

2n + 1, n ≥ 0 |

- {1, 4, 6, 8, 13, 12, 14, 24, 18, 20, 32, 24, 31, 40, 30, 32, 48, 48, 38, 56, 42, 44, 78, 48, 57, 72, 54, 72, 80, 60, 62, 104, 84, 68, 96, 72, 74, 124, 96, 80, 121, 84, 108, 120, ...}

## Example

In 1638, Descartes found the following "odd spoof perfect number" (*no other "odd spoof perfect number" has ever been found!*):

that is odd and perfect only if you suppose (incorrectly) that

is a "spoof-prime factor," giving the "spoof prime factorization"

for which the "freestyle sum of divisors" (i.e. the sum of divisors function where one is free to consider some composite factors as "spoof-prime factors") yields

## See also

- A033870 Divisors = 1 (mod 4) of Descartes's 198585576189.
- A033871 Divisors = 3 (mod 4) of Descartes's 198585576189.
- A058007 Freestyle perfect numbers (perfect numbers and spoof perfect numbers). (
*The only odd one known is 198585576189.*) - A174292 Spoof perfect numbers (freestyle perfect numbers which are not perfect numbers). (
*The only odd one known is 198585576189.*)

## Notes

- ↑ Anatomy of Integers, CRM Proceedings and Lecture Notes, Volume
**46**, 2008, p. 167.

## References

- Richard K. Guy,
*Unsolved Problems in Number Theory*(2004), p. 72. - Jean-Marie De Koninck,
*Ces nombres qui nous fascinent*, Ellipses Ed. (2008), p. 372. - Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip (2008). "Descartes numbers". in De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian.
*Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006*. CRM Proceedings and Lecture Notes.**46**. Providence, RI: American Mathematical Society. pp. 167–173. Zbl 1186.11004. ISBN 978-0-8218-4406-9.

## External links

- William D. Banks, Ahmet M. Güloğlu, C. Wesley Nevans, Filip Saidak, Descartes Numbers, 2008.
- C. Rivera (Ed.), Prime Puzzle 111. Spoof odd Perfect numbers, primepuzzles.net