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

1,1

Comments

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

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A243755, A243837.

Sequence in context: A297896 A217338 A217163 * A287119 A156846 A324711

Adjacent sequences: A243836 A243837 A243838 * A243840 A243841 A243842

Keyword

nonn

Author

Zhi-Wei Sun, Jun 12 2014

Status

approved