A291657 Primes p such that p is a primitive root modulo prime(p).

2, 3, 7, 11, 13, 41, 71, 79, 83, 107, 109, 131, 139, 157, 163, 173, 179, 191, 211, 223, 229, 263, 271, 277, 293, 311, 313, 317, 337, 353, 359, 367, 373, 389, 419, 431, 439, 449, 457, 463, 479, 521, 547, 569, 577, 593, 607, 641, 661, 709, 719, 727, 743, 757, 761, 769, 787, 811, 823, 827

Offset: 1

Author

Zhi-Wei Sun, Aug 28 2017

Keywords

• nonn

Comments

The conjecture in A291615 implies that the current sequence has infinitely many terms. In fact, if there are only finitely many primes p with p a primitive root modulo prime(p) and we let P denote the product of all such primes, then by Dirichlet's theorem there is a prime q == 1 (mod 4*P) and hence any prime p with p a primitive root modulo prime(p) is a quadratic residue modulo q and hence not a primitive root modulo q.

Conjecture: a(n)/(n*log(n)) has a positive limit as n tends to the infinity. Equivalently, all the terms in this sequence form a subset of the set of all primes with positive asymptotic density.

Examples

			a(1) = 2 since the first prime 2 is a primitive root modulo prime(2) = 3.
a(2) = 3 since the prime 3 is a primitive root modulo prime(3) = 5.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Crossrefs

Cf. A000040, A242345, A243164, A243403, A291615, A291690.

Programs

Mathematica

p[n_]:=p[n]=Prime[n];
n=0;Do[Do[If[Mod[p[k]^(Part[Divisors[p[p[k]]-1],i])-1,p[p[k]]]==0,Goto[aa]],{i,1,Length[Divisors[p[p[k]]-1]]-1}];
n=n+1;Print[n," ",p[k]];Label[aa],{k,1,145}]