I did a little test today looking for all the numbers such as their even divisors are exactly all of them a prime number minus 1, to verify possible properties of them. These are the first terms, it is not included at OEIS:
2, [2]
4, [2, 4]
6, [2, 6]
10, [2, 10]
12, [2, 4, 6, 12]
18, [2, 6, 18]
22, [2, 22]
30, [2, 6, 10, 30]
36, [2, 4, 6, 12, 18, 36]
46, [2, 46]
58, [2, 58]
I tried to look for the one with the longest list of even divisors, but it seems that the longest one is $36$, at least up to $10^6$:
$36$, even divisors $[2, 4, 6, 12, 18, 36]$, so the primes are $[3, 5, 7, 13, 19, 37]$.
For instance, for the same exercise for the even divisors being exactly all of them a prime number plus 1 (except $1$ in the case of the even divisor $2$) it seems to be $24$
$24$, $[2, 4, 6, 8, 12, 24]$, so the primes are $[3, 5, 7, 11, 23]$.
And for instance for the case in which both minus and plus one are a prime (or $1$ for the even divisor $2$) the longest one seems to be $12$: $[2, 4, 6, 12]$.
I would like to ask the following question:
These are heuristics, but I do not understand why it seems impossible to find a greater number than those small values such as all the even divisors comply with the property and that list of divisors is longer than the list of $36$. Is there a theoretical reason behind that or should it be possible to find a greater number (maybe very big) complying with the property? The way of calculating such possibility is related somehow with Diophantine equations?
Probably the reason is very simple, but I can not see it clearly. Thank you very much in advance!
