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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.
1
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jan 29 2021
STATUS
approved