|
%I #10 Jan 30 2021 04:07:05
%S 2,5,7,17,19,23,29,43,59,61,71,73,107,113,131,137,149,157,173,181,191,
%T 197,199,229,233,239,241,251,257,311,317,331,349,383,401,409,421,461,
%U 491,499,541,547,557,599,601,613,619,641,653,673,719,751,761,787,797,809
%N 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 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 Amiram Eldar, <a href="/A337986/b337986.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 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 2 is a term since A007814(A000166(k)) = A007814(k-1) for all k > 1.
%t 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 Cf. A000166, A000255, A007814, A068985.
%K nonn
%O 1,1
%A _Amiram Eldar_, Jan 29 2021
|
