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A Descartes number is defined as an odd number which would have been an odd perfect number, if one of its composite factors were prime. An example is:

$$ D = 3^2\times7^2\times11^2\times13^2\times22021, $$

for which the divisor sigma function equals $2D$ when $22021$ is assumed prime (which it is evidently not).

If one removes the requirement of being an odd number, one can find several examples of "even Descartes numbers". For instance, one observes that

$$ D_e = 3\times4\times5 = 60 $$

is such that $\sigma(D_e)=2D_e$ if one assumes that $4$ is prime. Indeed:

$$ \sigma(3\times5)\times(4+1) = 120 = 2D_e. $$

I have not found any mention of even Descartes numbers in literature. Are these interesting for any reason, have they been studied anywhere? Any references would be welcome.

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    $\begingroup$ Such numbers are certainly interesting, but in the case of odd numbers, there is a clear motivation to search for numbers somehow coming near to a perfect number because no single odd perfect number is known. $\endgroup$
    – Peter
    May 13, 2018 at 11:27
  • $\begingroup$ @Peter Thank you. Do you know any papers that have dealt with "even Descartes numbers"? Or any mention of them in literature? $\endgroup$
    – Klangen
    May 14, 2018 at 12:38
  • $\begingroup$ bump... so there are no such references? $\endgroup$
    – Klangen
    May 17, 2018 at 7:48
  • $\begingroup$ I have not checked this. But as said, in the case of even numbers, the motivation is small because the perfect numbers are completely classified by the Mersenne-primes. $\endgroup$
    – Peter
    May 17, 2018 at 8:01
  • $\begingroup$ @Peter Thank you! $\endgroup$
    – Klangen
    May 17, 2018 at 8:39

1 Answer 1

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Descartes numbers (from its original formulation in Banks, et. al's paper) are also known as spoof odd perfect numbers (as coined by Dittmer).

In your case, the correct search term to use is even spoof perfect number.

Google returns the following references:

https://oeis.org/wiki/Spoof_perfect_numbers#Even_spoof_perfect_numbers

Spoof odd perfect numbers by Dittmer

Problems and Puzzles on Spoof Perfect Numbers

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    $\begingroup$ You can also try searching for references related to the freestyle sum of divisors function, @PreservedFruit. $\endgroup$ May 20, 2018 at 12:13
  • $\begingroup$ Thank you, that's exactly what I was looking for! $\endgroup$
    – Klangen
    May 21, 2018 at 10:17
  • $\begingroup$ Quick follow-up question: do we know whether there are an infinity of spoof-perfect numbers (or odd spoof-perfect numbers)? I found no reference to the density of such numbers in any of the links. $\endgroup$
    – Klangen
    Nov 11, 2019 at 10:46
  • $\begingroup$ @Klangen: There is only one odd spoof-perfect number that is currently known, and you already have that number in your question. Many more even spoof-perfect numbers are known, but I do not know if we can prove that there are infu $\endgroup$ Nov 12, 2019 at 3:44
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    $\begingroup$ @Klangen: It is conjectured that there are only finitely many even completely spoof-perfect numbers, for a given number of composite factors. Check the last link in my answer. $\endgroup$ Nov 12, 2019 at 9:37

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