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%I #23 Aug 05 2019 03:03:01
%S 1,698,890,911,1003,1141,1413,1717,1807,1947,1948,2216,2254,2329,2455,
%T 2768,3169,3224,3537,3624,3737,3766,3896,3904,3921,3959,4027,4275,
%U 4359,4427,4649,4708,4845,5051,5378,5386,5396,5896,5897,6100,6223,6226,6351,6377
%N Positive integers n such that prime(n+i) is a primitive root modulo prime(n+j) for any distinct i and j among 0, 1, 2.
%C Conjecture: For any integer m > 0, there are infinitely many positive integers n such that prime(n+i) is a primitive root modulo prime(n+j) for any distinct i and j among 0, 1, ..., m.
%H Zhi-Wei Sun, <a href="/A243837/b243837.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 a(1) = 1 since prime(1) = 2 and prime(2) = 3 are primitive roots modulo prime(3) = 5, and 2 and 5 are primitive roots modulo 3, and 3 and 5 are primitive roots modulo 2.
%e a(2) = 698 since prime(698) = 5261 and prime(699) = 5273 are primitive roots modulo prime(700) = 5279, and 5261 and 5279 are primitive roots modulo 5273, and 5273 and 5279 are primitive roots modulo 5261.
%t dv[n_]:=Divisors[n]
%t m=0;Do[Do[If[Mod[Prime[n+1]^(Part[dv[Prime[n]-1],j]),Prime[n]]==1||Mod[Prime[n+2]^(Part[dv[Prime[n]-1],j]),Prime[n]]==1,Goto[aa]],{j,1,Length[dv[Prime[n]-1]]-1}];Do[If[Mod[Prime[n]^(Part[dv[Prime[n+1]-1],i]),Prime[n+1]]==1||Mod[Prime[n+2]^(Part[dv[Prime[n+1]-1],i]),Prime[n+1]]==1,Goto[aa]],{i,1,Length[dv[Prime[n+1]-1]]-1}];Do[If[Mod[Prime[n]^(Part[dv[Prime[n+2]-1],j]),Prime[n+2]]==1||Mod[Prime[n+1]^(Part[dv[Prime[n+2]-1],j]),Prime[n+2]]==1,Goto[aa]],{j,1,Length[dv[Prime[n+2]-1]]-1}];m=m+1;Print[m," ",n];Label[aa];Continue,{n,1,7990}]
%Y Cf. A000040, A243755, A243839.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jun 11 2014
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