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

1,1

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.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A233040 A233769 A038895 * A113244 A040152 A323353

Adjacent sequences: A291654 A291655 A291656 * A291658 A291659 A291660

Keyword

nonn

Author

Zhi-Wei Sun, Aug 28 2017

Status

approved