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A291657
Primes p such that p is a primitive root modulo prime(p).
3
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, 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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 28 2017
STATUS
approved