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A307627 Primes p such that 2 is a primitive root modulo p while 8 is not.

%I A307627 #17 Feb 16 2025 08:33:55
%S A307627 13,19,37,61,67,139,163,181,211,349,373,379,421,523,541,547,613,619,
%T A307627 661,709,757,787,829,853,859,877,883,907,1117,1123,1171,1213,1237,
%U A307627 1291,1381,1453,1483,1531,1549,1621,1669,1693,1741,1747,1861,1867,1987,2029,2053
%N A307627 Primes p such that 2 is a primitive root modulo p while 8 is not.
%C A307627 Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 3).
%C A307627 According to Artin's conjecture, the number of terms <= N is roughly ((2/5)*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).
%H A307627 Robert Israel, <a href="/A307627/b307627.txt">Table of n, a(n) for n = 1..10000</a>
%H A307627 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArtinsConstant.html">Artin's constant</a>
%H A307627 Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>
%e A307627 For p = 67, the multiplicative order of 2 modulo 67 is 66, while 8^22 == 2^(3*22) == 1 (mod 67), so 67 is a term.
%p A307627 select(p -> isprime(p) and numtheory:-order(2,p) = p-1,
%p A307627 [seq(i,i=1..10000,6)]); # _Robert Israel_, Apr 23 2019
%t A307627 Select[Prime@ Range[5, 310], And[FreeQ[#, 8], ! FreeQ[#, 2]] &@ PrimitiveRootList@ # &] (* _Michael De Vlieger_, Apr 23 2019 *)
%o A307627 (PARI) forprime(p=3, 2000, if(znorder(Mod(2, p))==(p-1) && p%3==1, print1(p, ", ")))
%Y A307627 Complement of A019338 with respect to A001122.
%Y A307627 Cf. also A005596, A000720, A307628.
%K A307627 nonn
%O A307627 1,1
%A A307627 _Jianing Song_, Apr 19 2019