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

1,1

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.

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.

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}.

Example

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

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]

Crossrefs

Cf. A000166, A000255, A007814, A068985.

Sequence in context: A306636 A101150 A276322 * A174281 A301916 A038875

Adjacent sequences: A337983 A337984 A337985 * A337987 A337988 A337989

Keyword

nonn

Author

Amiram Eldar, Jan 29 2021

Status

approved