A019338 Primes with primitive root 8.

3, 5, 11, 29, 53, 59, 83, 101, 107, 131, 149, 173, 179, 197, 227, 269, 293, 317, 347, 389, 419, 443, 461, 467, 491, 509, 557, 563, 587, 653, 659, 677, 701, 773, 797, 821, 827, 941, 947, 1019, 1061, 1091, 1109, 1187, 1229, 1259, 1277, 1283, 1301, 1307, 1373, 1427

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

To allow primes less than the specified primitive root m (here, 8) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 03 2019

Members of A001122 that are not congruent to 1 mod 3. - Robert Israel, Aug 12 2014

Terms greater than 3 are congruent to 5 or 11 modulo 24. - Jianing Song, May 12 2024 [Corrected on May 13 2025]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for primes by primitive root

Programs

Maple

select(t -> isprime(t) and numtheory:-order(8,t) = t-1, [2*i+1 $ i=1..1000]); # Robert Israel, Aug 12 2014

Mathematica

pr=8; Select[Prime[Range[200]], 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[#, 3],20]]==1 &]+1
(* Gerry Martens, Apr 28 2015 *)
Join[{3,5},Select[Prime[Range[250]],PrimitiveRoot[#,8]==8&]] (* Harvey P. Dale, Aug 10 2019 *)

PARI

is(n)=isprime(n) && n>2 && znorder(Mod(8,n))==n-1 \\ Charles R Greathouse IV, May 21 2015

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,3)==1. - Gerry Martens, May 15 2015

On Artin's conjecture, a(n) ~ (5/3A) n log n, where A = A005596 is Artin's constant. - Charles R Greathouse IV, May 21 2015