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.

2, 5, 7, 17, 19, 23, 29, 43, 59, 61, 71, 73, 107, 113, 131, 137, 149, 157, 173, 181, 191, 197, 199, 229, 233, 239, 241, 251, 257, 311, 317, 331, 349, 383, 401, 409, 421, 461, 491, 499, 541, 547, 557, 599, 601, 613, 619, 641, 653, 673, 719, 751, 761, 787, 797, 809

Offset: 1

Author

Amiram Eldar, Jan 29 2021

Keywords

• nonn

Comments

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.

Examples

			2 is a term since A007814(A000166(k)) = A007814(k-1) for all k > 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Piotr Miska, Arithmetic properties of the sequence of derangements, Journal of Number Theory, Vol. 163 (2016), pp. 114-145; arXiv preprint, arXiv:1508.01987 [math.NT], 2015.

Crossrefs

Cf. A000166, A000255, A007814, A068985.

Programs

Mathematica

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]

Formula

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}.