A243837 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.

1, 698, 890, 911, 1003, 1141, 1413, 1717, 1807, 1947, 1948, 2216, 2254, 2329, 2455, 2768, 3169, 3224, 3537, 3624, 3737, 3766, 3896, 3904, 3921, 3959, 4027, 4275, 4359, 4427, 4649, 4708, 4845, 5051, 5378, 5386, 5396, 5896, 5897, 6100, 6223, 6226, 6351, 6377

Offset

1,2

Comments

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.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Example

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.
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.

Mathematica

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

Crossrefs

Cf. A000040, A243755, A243839.

Sequence in context: A118059 A028500 A133251 * A116338 A293203 A157366

Adjacent sequences: A243834 A243835 A243836 * A243838 A243839 A243840

Keyword

nonn

Author

Zhi-Wei Sun, Jun 11 2014

Status

approved