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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 28 2017
STATUS
approved