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%I #40 Dec 03 2019 18:49:26
%S 3,5,11,29,53,59,83,101,107,131,149,173,179,197,227,269,293,317,347,
%T 389,419,443,461,467,491,509,557,563,587,653,659,677,701,773,797,821,
%U 827,941,947,1019,1061,1091,1109,1187,1229,1259,1277,1283,1301,1307,1373,1427
%N Primes with primitive root 8.
%C 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
%C Members of A001122 that are not congruent to 1 mod 3. - _Robert Israel_, Aug 12 2014
%H Vincenzo Librandi, <a href="/A019338/b019338.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>
%F 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
%F 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
%p select(t -> isprime(t) and numtheory:-order(8,t) = t-1, [2*i+1 $ i=1..1000]); # _Robert Israel_, Aug 12 2014
%t pr=8; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &] (* _N. J. A. Sloane_, Jun 01 2010 *)
%t a[p_, q_]:=Sum[2 Cos[2^n Pi/((2 q+1)(2 p+1))],{n,1,2 q p}]
%t 2 Select[Range[800],Rationalize[N[a[#, 3],20]]==1 &]+1
%t (* _Gerry Martens_, Apr 28 2015 *)
%t Join[{3,5},Select[Prime[Range[250]],PrimitiveRoot[#,8]==8&]] (* _Harvey P. Dale_, Aug 10 2019 *)
%o (PARI) is(n)=isprime(n) && n>2 && znorder(Mod(8,n))==n-1 \\ _Charles R Greathouse IV_, May 21 2015
%K nonn
%O 1,1
%A _David W. Wilson_
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