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Dec 29, 2020 at 17:16 vote accept T. Amdeberhan
Dec 29, 2020 at 17:04 comment added T. Amdeberhan @NoamD.Elkies: thanks for the references which I was aware of. I agree with your mod 3 comments. I also think that no such behavior occurs for other primes.
Dec 29, 2020 at 16:31 comment added Noam D. Elkies This is sequence 248 in the OEIS (oeis.org/A000248), which gives some further information including the exponential generating function $\exp(x \exp(x))$ and some references and further interpretations. Possibly one of those might lead to a proof. For what it's worth $c_2 = 3$ seems to be the only multiple of $3$, and I don't see a pattern in the valuation at $5$ and $7$.
Dec 28, 2020 at 20:48 answer added Fedor Petrov timeline score: 4
Dec 28, 2020 at 15:11 history asked T. Amdeberhan CC BY-SA 4.0