A307628 Primes p such that 2 is a primitive root modulo p while 32 is not.

11, 61, 101, 131, 181, 211, 421, 461, 491, 541, 661, 701, 821, 941, 1061, 1091, 1171, 1291, 1301, 1381, 1451, 1531, 1571, 1621, 1741, 1861, 1901, 1931, 2131, 2141, 2221, 2371, 2531, 2621, 2741, 2851, 2861, 3011, 3371, 3461, 3491, 3571, 3581, 3691, 3701, 3851, 3931

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 5).

By Artin's conjecture, the number of terms <= N is roughly ((4/19)*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 = 61, the multiplicative order of 2 modulo 61 is 60, while 32^12 == 2^(5*12) == 1 (mod 61), so 61 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 A019358 with respect to A001122.

Cf. also A005596, A000720, A307627.

Programs

Maple

select(p -> isprime(p) and numtheory:-order(2,p) = p-1,
[seq(i,i=1..10000,10)]); # Robert Israel, Apr 23 2019

Mathematica

{11}~Join~Select[Prime@ Range[11, 550], And[FreeQ[#, 32], ! FreeQ[#, 2]] &@ PrimitiveRootList@ # &] (* Michael De Vlieger, Apr 23 2019 *)

PARI

forprime(p=3, 4000, if(znorder(Mod(2, p))==(p-1) && p%5==1, print1(p, ", ")))