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

1,1

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

Links

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

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,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

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 *)

Prog

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

Crossrefs

Sequence in context: A109945 A045536 A319393 * A292539 A046134 A177932

Adjacent sequences: A019335 A019336 A019337 * A019339 A019340 A019341

Keyword

nonn

Author

David W. Wilson

Status

approved