A105880 Primes for which -8 is a primitive root.

5, 23, 29, 47, 53, 71, 101, 149, 167, 173, 191, 197, 239, 263, 269, 293, 311, 317, 359, 383, 389, 461, 479, 503, 509, 557, 599, 647, 653, 677, 701, 719, 743, 773, 797, 821, 839, 863, 887, 941, 983, 1031, 1061, 1109, 1151, 1223, 1229, 1277, 1301, 1319, 1367, 1373, 1439

Offset

1,1

Comments

From Jianing Song, May 12 2024: (Start)

Members of A105874 that are not congruent to 1 mod 3. Terms are congruent to 5 or 23 modulo 24.

According to Artin's conjecture, the number of terms <= N is roughly ((3/5)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N). (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Wikipedia, Artin's conjecture on primitive roots

Index entries for primes by primitive root

Formula

Let a(p,q)=sum(n=1,2*p*q,2*cos(2^n*Pi/((2*q+1)*(2*p+1)))). Then 2*p+1 is a prime of this sequence when a(p,9)==1. - Gerry Martens , May 21 2015

Mathematica

pr=-8; Select[Prime[Range[400]], MultiplicativeOrder[pr, # ] == #-1 &] (* N. J. A. Sloane, Jun 01 2010 *)
a[p_, q_]:= Sum[2 Cos[2^n Pi/((2 q+1)(2 p+1))], {n, 1, 2 q p}]
2 Select[Range[800], Rationalize[N[a[#, 9], 20]] == 1 &] + 1
(* Gerry Martens, Apr 28 2015 *)

Prog

(PARI) is(n)=isprime(n) && n>3 && znorder(Mod(-8, n))==n-1 \\ Charles R Greathouse IV, May 21 2015

Crossrefs

Cf. A105874, A005596, A000720.

Sequence in context: A243458 A067367 A140386 * A163587 A038922 A019367

Adjacent sequences: A105877 A105878 A105879 * A105881 A105882 A105883

Keyword

nonn

Author

N. J. A. Sloane, Apr 24 2005

Status

approved