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Descartes number

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."

Contents

Thus
 n
is an odd number[1]
${\displaystyle n=km,\quad k,\,m>1,\,m\nmid k,}$
where
 m
is taken as a "spoof-prime factor" which must not divide
 k
, such that
${\displaystyle \sigma (k)(m+1)=2n=2km,}$
where
 σ (k)
is the sum of divisors of
 k
.

Equivalently, we want

${\displaystyle {\frac {2k}{\sigma (k)}}={\frac {m+1}{m}}.}$
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!):

${\displaystyle 198585576189=3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}\cdot 19^{2}\cdot 61,\,}$

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

${\displaystyle 22021=19^{2}\cdot 61\,}$

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

${\displaystyle 198585576189=(3\cdot 7\cdot 11\cdot 13)^{2}\cdot 22021=3003^{2}\cdot 22021=9018009\cdot 22021,\,}$

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

{\displaystyle {\begin{aligned}\sigma _{\rm {freestyle}}(n)&={\bigg (}{\frac {3^{3}-1}{3-1}}{\bigg )}{\bigg (}{\frac {7^{3}-1}{7-1}}{\bigg )}{\bigg (}{\frac {{11}^{3}-1}{11-1}}{\bigg )}{\bigg (}{\frac {{13}^{3}-1}{13-1}}{\bigg )}(22021+1)={\bigg (}{\frac {26}{2}}{\bigg )}{\bigg (}{\frac {342}{6}}{\bigg )}{\bigg (}{\frac {1330}{10}}{\bigg )}{\bigg (}{\frac {2196}{12}}{\bigg )}(22022)=13\cdot 57\cdot 133\cdot 183\cdot 22022\\&=13\cdot (3\cdot 19)\cdot (7\cdot 19)\cdot (3\cdot 61)\cdot (2\cdot 7\cdot 11^{2}\cdot 13)=2\cdot (3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2})\cdot ({19}^{2}\cdot 61)=2\cdot (3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}\cdot 22021)=2n.\end{aligned}}\,}