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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 23 04:54 EDT 2020. Contains 337295 sequences. (Running on oeis4.)