A216838 Odd primes for which 2 is not a primitive root.

7, 17, 23, 31, 41, 43, 47, 71, 73, 79, 89, 97, 103, 109, 113, 127, 137, 151, 157, 167, 191, 193, 199, 223, 229, 233, 239, 241, 251, 257, 263, 271, 277, 281, 283, 307, 311, 313, 331, 337, 353, 359, 367, 383, 397, 401, 409, 431, 433, 439, 449, 457, 463, 479

Offset: 1

Author

V. Raman, Sep 17 2012

Keywords

• nonn

Comments

Alternately, for these primes p, the polynomial (x^p+1)/(x+1) is reducible over GF(2).

The prime p belongs to this sequence if and only if A002326((p-1)/2) != (p-1). If A002326((p-1)/2) = (p-1), then the prime p belongs to the sequence A001122. - V. Raman, Dec 01 2012

The only primitive root modulo 2 is 1. See A060749. Hence 2 should be added to this sequence in order to obtain the complement of A001122. - Wolfdieter Lang, May 19 2014

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Anand Bhardwaj, Luisa Degen, Radostin Petkov, and Sidney Stanbury, A Study of Cunningham Bounds through Rogue Primes, arXiv:2311.13375 [math.NT], 2023.

Crossrefs

Cf. A002326 (multiplicative order of 2 mod 2n+1)

Cf. A001122 (Primes for which 2 is a primitive root)

Cf. A115586 (Primes for which 2 is neither a primitive root nor a quadratic residue).

Programs

Maple

select(t -> isprime(t) and numtheory[order](2,t) <> t-1, [seq](2*i+1,i=1..1000)); # Robert Israel, May 20 2014

Mathematica

Select[Prime[Range[2, 100]], PrimitiveRoot[#] =!= 2 &] (* T. D. Noe, Sep 19 2012 *)

PARI

forprime(p=3, 1000, if(znorder(Mod(2,p))!=p-1, print(p)))

PARI

forprime(p=3, 1000, if(factormod((x^p+1)/(x+1), 2, 1)[1, 1]!=(p-1), print(p)))

Extensions

Name corrected by Wolfdieter Lang, May 19 2014