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

1,1

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).

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

Example

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.

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 *)

Prog

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

Crossrefs

Complement of A019358 with respect to A001122.

Cf. also A005596, A000720, A307627.

Sequence in context: A156704 A248874 A073625 * A106993 A066597 A199326

Adjacent sequences: A307625 A307626 A307627 * A307629 A307630 A307631

Keyword

nonn

Author

Jianing Song, Apr 19 2019

Status

approved