cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Jun 11 2014

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A000040, A243755, A243839.

Programs

  • 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}]

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.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Jun 12 2014

Keywords

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

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}]

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

Views

Author

Zhi-Wei Sun, Jun 12 2014

Keywords

Comments

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).

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Zhi-Wei Sun, Jun 14 2014

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A000040, A243755, A243837, A243839.

Programs

  • Mathematica
    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 *)
Showing 1-4 of 4 results.

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