A306144 Numbers k > 2 such that 3^(k-1) == 1 (mod k) and gcd(k, 2^(k-1)-1) = 1.

286, 16531, 24046, 49051, 72041, 182527, 192713, 232726, 258017, 327781, 442471, 443713, 453259, 574397, 625873, 652879, 655051, 668431, 705091, 903631, 1236031, 1241143, 1250833, 1287091, 1304446, 1309111, 1351601, 1414639, 1563151, 1817743, 1899451, 1908397

Offset: 1

Author

Thomas Ordowski, Aug 18 2018

Keywords

• nonn

Comments

The odd terms are "anti-Carmichael pseudoprimes (3,2)" defined as follows: numbers k > 1 such that 3^k == 3 (mod k) and gcd(k, 2^k-2) = 1. Cf. A300762 (2,3).

We impose k>2, since we want these to be pseudoprimes, thus composite numbers.

Crossrefs

Subsequence of A005935.

Cf. A130433.

Programs

Mathematica

Select[Range[3, 2*10^6], PowerMod[3, #-1, #] == 1 && GCD[#, #-1 + PowerMod[2, #-1, #]] == 1 &] (* Giovanni Resta, Aug 18 2018 *)

PARI

isok(k) = (k>2) && (Mod(3, k)^(k-1) == Mod(1, k)) && (gcd(k, 2^(k-1)-1) == 1); \\ Michel Marcus, Aug 18 2018

Extensions

More terms from Michel Marcus, Aug 18 2018

Further terms from Giovanni Resta, Aug 18 2018