A316907 Numbers k such that 2^(k-1) == 1 (mod k) and p-1 does not divide k-1 for every prime p dividing k.

7957, 23377, 35333, 42799, 49981, 60787, 129889, 150851, 162193, 164737, 241001, 249841, 253241, 256999, 280601, 318361, 452051, 481573, 556169, 580337, 617093, 665333, 722201, 838861, 877099, 1016801, 1251949, 1252697, 1325843, 1507963, 1534541, 1678541, 1826203, 1969417

Offset: 1

Author

Thomas Ordowski, Jul 16 2018

Keywords

• nonn

Comments

Numbers k such that 2^k == 2 (mod k) and gcd(k,b^k-b) = 1 for some b > 2.

Problem: are there infinitely many such "anti-Carmichael pseudoprimes"?

All semiprime terms of A316906 are in the sequence; i.e., numbers m in A214305 such that p-1 does not divide q-1, where m = pq and p < q are primes.

Examples

			7957 = 73*109 is pseudoprime and 72 does not divide 7956 (of course also 108 does not divide 7956), note that 72 does not divide 108.
617093 = 43*113*127 is the smallest such pseudoprime with more than two prime factors.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A001567 and of A316906.

Cf. A121707 (probably "anti-Carmichael numbers": n such that p-1 does not divide n-1 for every prime p dividing n).

Programs

Mathematica

Select[Select[Range[2*10^6], PowerMod[2, # - 1, #] == 1 &], Function[k, AllTrue[FactorInteger[k][[All, 1]] - 1, Mod[k - 1, #] != 0 &]]] (* Michael De Vlieger, Jul 20 2018 *)

Extensions

a(7)-a(34) from Michel Marcus, Jul 16 2018