A243839 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.

8560, 9719, 19228, 20509, 32117, 32352, 44512, 48086, 56967, 63104, 72233, 72538, 73481, 84831, 85736, 87999, 89747, 98220, 102116, 108246, 116228, 123982, 141709, 144344, 147685, 148099, 171214, 173916, 177322, 180836

Offset: 1

Author

Zhi-Wei Sun, Jun 12 2014

Keywords

• nonn

Comments

According to the general conjecture in A243837, this sequence should have infinitely many terms.

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..50
• Hao Pan, Z.-W. Sun, Consecutive primes and Legendre symbols, arXiv preprint arXiv:1406.5951 [math.NT], 2014-2018.

Crossrefs

Cf. A000040, A243755, A243837.

Programs

Mathematica

dv[n_]:=Divisors[n]
p[n_]:=Prime[n]
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}]