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

13, 19, 37, 61, 67, 139, 163, 181, 211, 349, 373, 379, 421, 523, 541, 547, 613, 619, 661, 709, 757, 787, 829, 853, 859, 877, 883, 907, 1117, 1123, 1171, 1213, 1237, 1291, 1381, 1453, 1483, 1531, 1549, 1621, 1669, 1693, 1741, 1747, 1861, 1867, 1987, 2029, 2053

Offset: 1

Author

Jianing Song, Apr 19 2019

Keywords

• nonn

Comments

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

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).

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Artin's constant
• Wikipedia, Artin's conjecture on primitive roots

Crossrefs

Complement of A019338 with respect to A001122.

Cf. also A005596, A000720, A307628.

Programs

Maple

select(p -> isprime(p) and numtheory:-order(2,p) = p-1,
[seq(i,i=1..10000,6)]); # Robert Israel, Apr 23 2019

Mathematica

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

PARI

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