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A307628 Primes p such that 2 is a primitive root modulo p while 32 is not. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 9 22:27 EST 2019. Contains 329880 sequences. (Running on oeis4.)