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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."

Thus 
n
is an odd number[1]
where 
m
is taken as a "spoof-prime factor" which must not divide 
k
, such that
where 
σ (k)
is the sum of divisors of 
k
.

Equivalently, we want

A008438 Sum of divisors of 
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

Notes

  1. 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