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A019338 Primes with primitive root 8. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 1 13:00 EST 2021. Contains 349429 sequences. (Running on oeis4.)