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A337986 Prime numbers p such that v_p(A000166(k)) = v_p(k-1) for all k > 1, where v_p(k) is the p-adic valuation of k.

%I A337986 #10 Jan 30 2021 04:07:05
%S A337986 2,5,7,17,19,23,29,43,59,61,71,73,107,113,131,137,149,157,173,181,191,
%T A337986 197,199,229,233,239,241,251,257,311,317,331,349,383,401,409,421,461,
%U A337986 491,499,541,547,557,599,601,613,619,641,653,673,719,751,761,787,797,809
%N A337986 Prime numbers p such that v_p(A000166(k)) = v_p(k-1) for all k > 1, where v_p(k) is the p-adic valuation of k.
%C A337986 Miska (2016) proved that the complement of this sequence within the primes is infinite, and conjectured that this sequence is also infinite, and that its asymptotic density within the primes is 1/e (A068985). Numerically, he found that there are 28990 terms below 10^6, which are about 37% of all the primes less than 10^6.
%H A337986 Amiram Eldar, <a href="/A337986/b337986.txt">Table of n, a(n) for n = 1..1000</a>
%H A337986 Piotr Miska, <a href="https://doi.org/10.1016/j.jnt.2015.11.014">Arithmetic properties of the sequence of derangements</a>, Journal of Number Theory, Vol. 163 (2016), pp. 114-145; <a href="https://arxiv.org/abs/1508.01987">arXiv preprint</a>, arXiv:1508.01987 [math.NT], 2015.
%F A337986 A prime p is a term if and only if p does not divide any of the numbers A000255(k), k in {2, ..., p+1}.
%e A337986 2 is a term since A007814(A000166(k)) = A007814(k-1) for all k > 1.
%t A337986 e[n_] := e[n] = Subfactorial[n]/(n - 1); q[p_] := PrimeQ[p] && AllTrue[Table[e[n], {n, 2, p + 1}], ! Divisible[#, p] &]; Select[Range[1000], q]
%Y A337986 Cf. A000166, A000255, A007814, A068985.
%K A337986 nonn
%O A337986 1,1
%A A337986 _Amiram Eldar_, Jan 29 2021