A231370 Squarefree composite numbers k such that 2 is a primitive root for all prime factors of k.

15, 33, 39, 55, 57, 65, 87, 95, 111, 143, 145, 159, 165, 177, 183, 185, 195, 201, 209, 247, 249, 265, 285, 295, 303, 305, 319, 321, 335, 377, 393, 407, 415, 417, 429, 435, 447, 481, 489, 505, 519, 535, 537, 543, 551, 555, 583, 591, 627, 633, 649, 655, 671, 681

Offset: 1

Author

Arkadiusz Wesolowski, Nov 08 2013

Keywords

• nonn

Comments

If k is the smallest integer satisfying 10^k == 1 (mod p), we say that 10 has order k (mod p). If n is the product of distinct primes p_i, the period of 1/n in base b is the least common multiple of the orders of b (mod p_i), provided b and n are relatively prime.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Primitive Root.
• Wikipedia, Binary number.

Crossrefs

Subsequence of A024556.

Cf. A001122, A231371, A231372.

Programs

Mathematica

q[n_] := CompositeQ[n] && SquareFreeQ[n] && AllTrue[FactorInteger[n][[;;,1]],  MultiplicativeOrder[2, #] == # - 1 &]; Select[Range[700], q] (* Amiram Eldar, Oct 03 2021 *)

PARI

isok(k) = if ((k>1) && (k%2) && !isprime(k) && issquarefree(k), my(f=factor(k)[,1]~); for (j=1, #f, if (znorder(Mod(2, f[j])) != (f[j]-1), return(0))); return (1)); return (0); \\ Michel Marcus, Oct 03 2021