A223036 -id:A223036 - cp's OEIS frontend

A047936 Primes whose smallest positive primitive root (A001918) is not prime.

2, 41, 109, 151, 229, 251, 271, 313, 337, 367, 409, 439, 733, 761, 971, 991, 1021, 1031, 1069, 1289, 1297, 1303, 1429, 1471, 1489, 1759, 1783, 1789, 1811, 1871, 1873, 1879, 2137, 2411, 2441, 2551, 2749, 2791, 2971, 3001, 3061, 3079, 3109, 3221, 3229

Offset: 1

Felice Russo

Subsequence of A222717 = primes whose smallest positive quadratic nonresidue is not a primitive root. (Proof. If p is not in A222717, then the smallest positive quadratic nonresidue of p is a primitive root g. Since the smallest positive quadratic nonresidue is always a prime, g is prime. But since all primitive roots are quadratic nonresidues, g is the smallest positive primitive root of p. Hence p is not in A047936.) - Jonathan Sondow, Mar 13 2013.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes by primitive root

Cf. A222717, A223036.

lst={}; Do[p=Prime[n]; pr=PrimitiveRoot[p]; If[pr>1&&!PrimeQ[pr], AppendTo[lst, p]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 24 2009 *)
Select[Prime[Range[500]],!PrimeQ[PrimitiveRoot[#]]&] (* Harvey P. Dale, Oct 24 2011 *)

select(p->!isprime(lift(znprimroot(p))),primes(999)) \\ reverse order of arguments if using an old version of GP
\\ Charles R Greathouse IV, Oct 24 2011

More terms from James Sellers, Dec 22 1999

A222717 Primes p whose smallest positive quadratic nonresidue is not a primitive root of p.

Original entry on oeis.org

2, 41, 43, 103, 109, 151, 157, 191, 229, 251, 271, 277, 283, 307, 311, 313, 331, 337, 367, 397, 409, 439, 457, 499, 571, 643, 683, 691, 727, 733, 739, 761, 769, 811, 911, 919, 967, 971, 991, 997, 1013, 1021, 1031, 1051, 1069, 1093, 1151, 1163, 1181, 1289

Offset: 1

Jonathan Sondow, Mar 12 2013

nonn

Same as primes p such that if q is the smallest positive quadratic nonresidue mod p, then either q == 0 mod p or q^k == 1 mod p for some positive integer k < p-1.
A primitive root of an odd prime p is always a quadratic nonresidue mod p. (Proof. If g == x^2 mod p, then g^((p-1)/2) == x^(p-1) == 1 mod p, and so g is not a primitive root of p.) But a quadratic nonresidue mod p may or may not be a primitive root of p.
Supersequence of A047936 = primes whose smallest positive primitive root is not prime. (Proof. If p is not in A222717, then the smallest positive quadratic nonresidue of p is a primitive root g. Since the smallest positive quadratic nonresidue is always a prime, g is prime. But since all primitive roots are quadratic nonresidues, g is the smallest positive primitive root of p. Hence p is not in A047936.)
See A001918 (least positive primitive root of the n-th prime) and  A053760 (smallest positive quadratic nonresidue of the n-th prime) for references and additional comments and links.

			The smallest positive quadratic nonresidue of 2 is 2 itself, and 2 is not a primitive root of 2, so 2 is a member.
The smallest positive quadratic nonresidue of 41 is 3, and 3 is not a primitive root of 41, so 41 is a member.

Index entries for primes by primitive root

Cf. A001918, A047936, A053760, A223036.

nn = 300; NR = (Table[p = Prime[n]; First[ Select[ Range[p], JacobiSymbol[#, p] != 1 &]], {n, nn}]); Select[ Prime[ Range[nn]], Mod[ NR[[PrimePi[#]]], #] == 0 || MultiplicativeOrder[ NR[[PrimePi[#]]], #] < # - 1 &]

cp's OEIS Frontend

A047936 Primes whose smallest positive primitive root (A001918) is not prime.

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