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.
Best Answer
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