A225185 Primes p which do not have a primitive root that divides p+1.

7, 23, 31, 43, 47, 71, 73, 79, 103, 127, 151, 157, 167, 191, 193, 199, 223, 239, 241, 263, 271, 277, 283, 311, 313, 331, 337, 359, 367, 383, 397, 409, 431, 439, 457, 463, 479, 487, 503, 571, 577, 599, 607, 631, 647, 673, 691, 719, 727, 733, 739, 743, 751, 811, 823, 839, 863, 887, 911, 919, 967, 983, 991, 997

Offset: 1

Author

N. J. A. Sloane, May 04 2013

Keywords

• nonn

Examples

			The primitive roots modulo 97 are 5, 7, 10, 13, 14, 15, 17, 21, 23, 26, 29, 37, 38, 39, ..., and 7 divides 98, so 97 is not a term of this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Arto Lepistö, Francesco Pappalardi and Kalle Saari, Transposition Invariant Words, Theoret. Comput. Sci., Vol. 380, No. 3 (2007), pp. 377-387.

Crossrefs

Cf. A060749, A225184 (complement), A001122.

Programs

Magma

[p: p in PrimesUpTo(1000) | not exists{r: r in [1..p-1] | IsPrimitive(r,p) and IsZero((p+1) mod r)}]; // Bruno Berselli, May 10 2013

Mathematica

q[n_] := PrimeQ[n] && AllTrue[PrimitiveRootList[n], ! Divisible[n + 1, #] &]; Select[Range[1000], q] (* Amiram Eldar, Oct 07 2021 *)
Select[Prime[Range[200]],NoneTrue[(#+1)/PrimitiveRootList[#],IntegerQ]&] (* Harvey P. Dale, Sep 08 2024 *)

PARI

forprime(p=2,1000, i=0;fordiv(p+1,X, if(znorder(Mod(X,p))==eulerphi(p), i=1)); if(i==0,print1(p", "))) \\ V. Raman, May 04 2012

Extensions

More terms from V. Raman, May 04 2013