cp's OEIS Frontend

A291615 Number of primes p < prime(n) such that p is a primitive root modulo prime(n) and also a primitive root modulo prime(p).

%I A291615 #10 Aug 28 2017 00:47:05
%S A291615 0,1,2,1,2,3,3,3,2,3,3,2,3,1,3,3,3,2,5,3,2,2,4,5,5,5,2,3,3,3,4,2,6,3,
%T A291615 11,4,3,8,9,8,10,7,6,3,9,6,6,6,11,10,11,9,9,9,12,11,13,3,6,10,7,15,5,
%U A291615 6,7,13,7,8,14,10,13,19,12,14,11,18,15,11,15,8
%N A291615 Number of primes p < prime(n) such that p is a primitive root modulo prime(n) and also a primitive root modulo prime(p).
%C A291615 Conjecture: a(n) > 0 for all n > 1. In other words, for any odd prime p there is a prime q < p such that q is a primitive root modulo p and also a primitive root modulo prime(q).
%C A291615 According to page 377 in Guy's book, P. Erdős asked whether for any sufficiently large prime p there exists a prime q < p which is a primitive root modulo p.
%D A291615 R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, New York, 2004.
%H A291615 Zhi-Wei Sun, <a href="/A291615/b291615.txt">Table of n, a(n) for n = 1..2000</a>
%H A291615 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv:1405.0290 [math.NT], 2014.
%e A291615 a(2) = 1 since the prime 2 < prime(2) = 3 is a primitive root modulo prime(2) = 3.
%e A291615 a(4) = 1 since the prime 3 < prime(4) = 7 is a primitive root modulo prime(4) = 7 and also a primitive root modulo prime(3) = 5.
%e A291615 a(14) = 1 since the prime 3 < prime(14) = 43 is a primitive root modulo prime(14) = 43 and also a primitive root modulo prime(3) = 5.
%t A291615 rMod[m_,n_]:=rMod[m,n]=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
%t A291615 p[n_]:=p[n]=Prime[n];
%t A291615 Do[r=0;Do[Do[If[Mod[p[g]^(Part[Divisors[p[n]-1],i])-1,p[n]]==0,Goto[aa]],{i,1,Length[Divisors[p[n]-1]]-1}];
%t A291615 Do[If[Mod[p[g]^(Part[Divisors[p[p[g]]-1],j])-1,p[p[g]]]==0,Goto[aa]],{j,1,Length[Divisors[p[p[g]]-1]]-1}];
%t A291615 r=r+1;Label[aa],{g,1,n-1}];Print[n," ",r],{n,1,80}]
%Y A291615 Cf. A000040, A242345, A243164, A243403.
%K A291615 nonn
%O A291615 1,3
%A A291615 _Zhi-Wei Sun_, Aug 27 2017