A243837 -id:A243837 - cp's OEIS frontend

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

Zhi-Wei Sun, Jun 12 2014

nonn

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

			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.

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.

Cf. A000040, A243755, A243837.

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}]

A243847 a(n) = |{0 < k < n: prime(k) is a primitive root modulo prime(n) and also a primitive root modulo prime(2*n)}|.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 1, 5, 2, 1, 1, 3, 5, 2, 3, 2, 3, 5, 4, 4, 7, 1, 5, 5, 7, 7, 6, 8, 6, 6, 5, 6, 3, 5, 4, 8, 6, 4, 5, 6, 6, 12, 8, 15, 17, 7, 10, 8, 11, 10, 8, 9, 10, 7, 18, 6, 15, 4, 9, 5, 10, 10, 8

Offset: 1

Zhi-Wei Sun, Jun 12 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 2.

(ii) For any integer n > 4, there is a primitive root 0 < g < prime(n) modulo prime(n) which is also a primitive root modulo prime(n+1).

			a(3) = 1 since prime(1) = 2 is a primitive root modulo prime(3) = 5 and also a primitive root modulo prime(2*3) = 13. Note that prime(2) = 3 is not a primitive root modulo prime(2*3) = 13 since 3^3 == 1 (mod 13).

Zhi-Wei Sun, Table of n, a(n) for n = 1..6000
Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A242345, A243755, A243837, A243839.

dv[n_]:=Divisors[n]
Do[m=0;Do[Do[If[Mod[(Prime[k])^(Part[dv[Prime[n]-1],i]),Prime[n]]==1,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}]; Do[If[Mod[(Prime[k])^(Part[dv[Prime[2n]-1],j]),Prime[2n]]==1,Goto[aa]],{j,1,Length[dv[Prime[2n]-1]]-1}];m=m+1;Label[aa];Continue,{k,1,n-1}];Print[n," ",m];Continue,{n,1,70}]

A243901 Positive integers n such that p_{n+i} is a quadratic residue modulo p_{n+j} for any distinct i and j among 0, 1, ..., 6.

Original entry on oeis.org

178633, 2355662, 4892172, 5702347, 9256159, 9572343, 13837265, 15147032, 15429648, 15822376, 16603935, 20925043, 22128672, 22462201, 22689295, 27145167, 28031877, 28470899, 29246422, 30772941, 31211796, 32372758

Offset: 1

Zhi-Wei Sun, Jun 14 2014

nonn

Conjecture: For any integer m > 0, there are infinitely many positive integers n such that p_{n+i} is a quadratic residue modulo p_{n+j} for any distinct i and j among 0, 1, ..., m.

			a(1) = 178633 since any 6 primes among the 7 integers prime(178633) = 2434589, prime(178634) = 2434609, prime(178635) = 2434613, prime(178636) = 2434657, prime(178637) = 2434669, prime(178638) = 2434673 and prime(178639) = 2434681 are quadratic residues modulo the remaining one of the 7 primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Hao Pan, Z.-W. Sun, Consecutive primes and Legendre symbols, arXiv preprint arXiv:1406.5951 [math.NT], 2014-2018.

Cf. A000040, A243755, A243837, A243839.

q[i_,j_]:=JacobiSymbol[Prime[i],Prime[j]]
m=0;Do[Do[If[q[n+i,n+j]==-1,Goto[aa]],{i,0,6},{j,0,6}]; m=m+1;Print[m," ",n];Label[aa];Continue,{n,1,32372758}]
Reap[ Do[ If[ Catch[ Do[ If[ JacobiSymbol[Prime[n + i], Prime[n + j]] != 1, Throw@False], {i, 0, 5}, {j, i + 1, 6}]; True], Sow[n]], {n, 32372758}]][[2, 1]] (* Michael Somos, Jun 15 2014 *)

cp's OEIS Frontend

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.

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A243847 a(n) = |{0 < k < n: prime(k) is a primitive root modulo prime(n) and also a primitive root modulo prime(2*n)}|.

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A243901 Positive integers n such that p_{n+i} is a quadratic residue modulo p_{n+j} for any distinct i and j among 0, 1, ..., 6.

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