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

1,1

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.

Links

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

Example

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.

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 *)

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).

Sequence in context: A064246 A260959 A316906 * A293623 A316908 A254138

Adjacent sequences: A316904 A316905 A316906 * A316908 A316909 A316910

Keyword

nonn

Author

Thomas Ordowski, Jul 16 2018

Extensions

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

Status

approved