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.

Previous Showing 11-18 of 18 results.

A229848 Values of n such that the equation x^2 - 3*n*y^2 = n has integer solutions.

Original entry on oeis.org

1, 4, 7, 9, 13, 16, 19, 25, 28, 31, 36, 37, 43, 49, 52, 61, 63, 64, 67, 76, 79, 81, 91, 100, 103, 109, 112, 117, 121, 124, 127, 139, 144, 148, 151, 157, 163, 169, 171, 172, 175, 181, 193, 196, 199, 208, 211, 217, 223, 225, 229, 244, 247, 252, 256, 268, 271
Offset: 1

Views

Author

Colin Barker, Oct 07 2013

Keywords

Examples

			37 appears in the sequence because the equation x^2 - 111*y^2 = 37 has integer solutions.

Crossrefs

Cf. A172000, A227939

Programs

  • Mathematica
    Select[Range[300],Length[FullSimplify[Solve[x^2-3*#*y^2==#,{x,y},Integers]/.C[1]->1]]>0&] (* Vaclav Kotesovec, Oct 08 2013 *)

A197169 Values of gcd(n,y) for successive y = A197128(n).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 46, 2, 2, 2, 2, 3, 10, 6, 2, 3, 3, 2, 2, 2, 6, 5, 2, 2, 5, 2, 2, 2, 2, 2, 2, 22, 5, 2, 2, 3, 2, 2, 3, 2, 2, 3, 46, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 14, 2, 4, 3, 2, 2, 2, 4, 14, 3, 2, 3, 2, 5, 2, 2, 2, 5, 2, 2, 2, 3, 6, 29, 3, 2, 2, 3
Offset: 1

Views

Author

Artur Jasinski, Oct 11 2011

Keywords

Crossrefs

Cf. A087643, A172000, A194366, A197115, A197127, A197128.

Programs

  • Mathematica
    cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[cr, GCD[d4, n]]]], {n, 2, 20000}]; cr

A197171 Values k such that singular quadratic unity of Q(k) have gcd(k,y) = 2.

Original entry on oeis.org

6, 14, 22, 30, 34, 38, 42, 54, 56, 62, 66, 86, 94, 102, 110, 118, 126, 132, 134, 138, 142, 146, 150, 156, 158, 166, 174, 178, 182, 186, 190, 194, 198, 206, 210, 214, 220, 222, 228, 230, 246, 254, 258, 262, 270, 278, 282, 286, 294, 302, 306, 310, 322, 326
Offset: 2

Views

Author

Artur Jasinski, Oct 11 2011

Keywords

Comments

Conjecture: This sequence is infinite.

Crossrefs

Cf. A087643, A172000, A194366, A197115, A197127, A197128, A197170.

Programs

  • Mathematica
    cr = {}; ck = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[ck, GCD[d4, n]]; AppendTo[cr, n]]], {n, 2, 200000}];aa = {}; Do[If[ck[[n]] == 2, AppendTo[aa, cr[[n]]]], {n, 1, Length[cr]}]; aa

A230108 Values of d such that the equation x^2 - d*y^2 = 2*d has integer solutions.

Original entry on oeis.org

2, 3, 6, 8, 11, 12, 18, 19, 22, 24, 27, 32, 38, 43, 44, 48, 50, 51, 54, 59, 66, 67, 72, 75, 76, 83, 86, 88, 96, 98, 99, 102, 107, 108, 114, 118, 123, 128, 131, 134, 139, 146, 147, 150, 152, 162, 163, 166, 171, 172, 176, 178, 179
Offset: 1

Views

Author

Colin Barker, Oct 09 2013

Keywords

Examples

			43 appears in the sequence because the equation x^2 - 43*y^2 = 86 has integer solutions, such as (x,y) = (387,59).

Crossrefs

Cf. A172000, A230109.

Programs

  • Mathematica
    Select[Range[200],FindInstance[x^2-#*y^2==2*#,{x,y},Integers]!={}&] (* Harvey P. Dale, Jun 22 2019 *)
  • PARI
    is(n)=sol=bnfisintnorm(bnfinit(z^2-n),2*n);if(!#sol,0,p=polcoeff(sol[1],0);p==floor(p)) \\ Ralf Stephan, Oct 19 2013

A230109 Values of d such that the equation x^2 - d*y^2 = 3*d has integer solutions.

Original entry on oeis.org

3, 7, 12, 13, 19, 21, 27, 28, 31, 39, 43, 48, 52, 57, 61, 63, 67, 73, 75, 76, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 124, 127, 129, 133, 139, 147, 151, 156, 157, 163, 171, 172, 175
Offset: 1

Views

Author

Colin Barker, Oct 09 2013

Keywords

Examples

			43 appears in the sequence because the equation x^2 - 43*y^2 = 129 has integer solutions, such as (x,y) = (86,13).

Crossrefs

Cf. A172000, A230108.

A197120 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))has norm -1 and minimum one from two parts of fundamental unit are not integer.

Original entry on oeis.org

5, 13, 20, 29, 45, 52, 53, 61, 80, 85, 109, 116, 117, 125, 149, 157, 173, 180, 181, 208, 212, 229, 244, 245, 261, 277, 293, 317, 320, 325, 340, 365, 397, 405, 421, 436, 445, 461, 464, 468, 477, 493, 500, 509, 533, 541, 549, 565, 596, 605, 613, 628, 629, 637
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2011

Keywords

Comments

Numbers that occur in A172000 and not in A197115.

Crossrefs

Cf. A087643, A172000, A194366, A197115.

Programs

  • Mathematica
    cr = {}; Do[ If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d3 = Expand[(d1 + d2) (d1 - d2)]; If[d3 == -1, k1 = Max[Denominator[d1], Denominator[d2]]; If[k1 == 1, , AppendTo[cr, n]]]], {n, 2, 2000}]; cr

A197121 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))has norm 1 and minimum one from two parts of fundamental unit are not integer.

Original entry on oeis.org

21, 69, 77, 84, 93, 133, 165, 189, 205, 213, 221, 237, 253, 276, 285, 301, 308, 309, 336, 341, 357, 372, 413, 429, 437, 453, 469, 501, 517, 525, 532, 581, 589, 597, 621, 645, 660, 669, 693, 717, 741, 749, 756, 789, 805, 820, 837, 852, 861, 869, 884, 893, 917
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2011

Keywords

Comments

Numbers that occur in A087643 and not in A194366.

Crossrefs

Cf. A087643, A172000, A194366, A197115.

Programs

  • Mathematica
    cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d3 = Expand[(d1 + d2) (d1 - d2)]; If[d3 == 1, k1 = Max[Denominator[d1], Denominator[d2]]; If[k1 == 1, , AppendTo[cr, n]]]], {n, 2, 2000}]; cr

A197172 Values k such that singular quadratic unity of Q(k) have gcd(k, y) = 3.

Original entry on oeis.org

69, 87, 93, 159, 177, 183, 249, 267, 276, 312, 321, 327, 348, 372, 387, 417, 471, 597, 633, 636, 699, 711, 717, 723, 741, 747, 831, 849, 879, 921, 927, 987, 993, 1005, 1068, 1104, 1137, 1179, 1248, 1251, 1272, 1293, 1299, 1317, 1320, 1353, 1359, 1383, 1392
Offset: 2

Views

Author

Artur Jasinski, Oct 11 2011

Keywords

Crossrefs

Cf. A087643, A172000, A194366, A197115, A197127, A197128, A197170, A197171.

Programs

  • Mathematica
    cr = {}; ck = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[ck, GCD[d4, n]]; AppendTo[cr, n]]], {n, 2, 200000}]; aa = {}; Do[If[ck[[n]] == 3, AppendTo[aa, cr[[n]]]], {n, 1, Length[cr]}]; aa
Previous Showing 11-18 of 18 results.

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