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%I #26 Aug 05 2019 02:01:26
%S 8560,9719,19228,20509,32117,32352,44512,48086,56967,63104,72233,
%T 72538,73481,84831,85736,87999,89747,98220,102116,108246,116228,
%U 123982,141709,144344,147685,148099,171214,173916,177322,180836
%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, 3.
%C According to the general conjecture in A243837, this sequence should have infinitely many terms.
%H Zhi-Wei Sun, <a href="/A243839/b243839.txt">Table of n, a(n) for n = 1..50</a>
%H Hao Pan, Z.-W. Sun, <a href="http://arxiv.org/abs/1406.5951">Consecutive primes and Legendre symbols</a>, arXiv preprint arXiv:1406.5951 [math.NT], 2014-2018.
%e a(1) = 8560 since prime(8560) = 88259, prime(8561) = 88261, and prime(8562) = 88289 are primitive roots modulo prime(8563) = 88301, and 88259, 88261, 88301 are primitive roots modulo 88289, and 88259, 88289, 88301 are primitive roots modulo 88261, and 88261, 88289, 88301 are primitive roots modulo 88259.
%t dv[n_]:=Divisors[n]
%t p[n_]:=Prime[n]
%t m=0; Do[Do[If[Mod[p[n]^(Part[dv[p[n+3]-1],i]),p[n+3]]==1||Mod[p[n+1]^(Part[dv[p[n+3]-1],i]), p[n+3]]==1||Mod[p[n+2]^(Part[dv[p[n+3]-1],i]),p[n+3]]==1,Goto[aa]],{i,1,Length[dv[p[n+3]-1]]-1}]; Do[If[Mod[p[n]^(Part[dv[p[n+2]-1],i]),p[n+2]]==1||Mod[p[n+1]^(Part[dv[p[n+2]-1],i]), p[n+2]]==1||Mod[p[n+3]^(Part[dv[p[n+2]-1],i]),p[n+2]]==1,Goto[aa]],{i,1,Length[dv[p[n+2]-1]]-1}]; Do[If[Mod[p[n]^(Part[dv[p[n+1]-1],i]),p[n+1]]==1||Mod[p[n+2]^(Part[dv[p[n+1]-1],i]), p[n+1]]==1||Mod[p[n+3]^(Part[dv[p[n+1]-1],i]),p[n+1]]==1,Goto[aa]],{i,1,Length[dv[p[n+1]-1]]-1}]; Do[If[Mod[p[n+1]^(Part[dv[p[n]-1],i]),p[n]]==1||Mod[p[n+2]^(Part[dv[p[n]-1],i]), p[n]]==1||Mod[p[n+3]^(Part[dv[p[n]-1],i]),p[n]]==1,Goto[aa]],{i,1,Length[dv[p[n]-1]]-1}]; m=m+1;Print[m," ",n];Label[aa];Continue,{n,1,108246}]
%Y Cf. A000040, A243755, A243837.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jun 12 2014
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