A307627 - OEIS

%I #17 Feb 16 2025 08:33:55

%S 13,19,37,61,67,139,163,181,211,349,373,379,421,523,541,547,613,619,

%T 661,709,757,787,829,853,859,877,883,907,1117,1123,1171,1213,1237,

%U 1291,1381,1453,1483,1531,1549,1621,1669,1693,1741,1747,1861,1867,1987,2029,2053

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

%C Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 3).

%C 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 Robert Israel, <a href="/A307627/b307627.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArtinsConstant.html">Artin's constant</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>

%e 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 select(p -> isprime(p) and numtheory:-order(2,p) = p-1,

%p [seq(i,i=1..10000,6)]); # _Robert Israel_, Apr 23 2019

%t Select[Prime@ Range[5, 310], And[FreeQ[#, 8], ! FreeQ[#, 2]] &@ PrimitiveRootList@ # &] (* _Michael De Vlieger_, Apr 23 2019 *)

%o (PARI) forprime(p=3, 2000, if(znorder(Mod(2, p))==(p-1) && p%3==1, print1(p, ", ")))

%Y Complement of A019338 with respect to A001122.

%Y Cf. also A005596, A000720, A307628.

%K nonn,changed

%O 1,1

%A _Jianing Song_, Apr 19 2019