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 1-9 of 9 results.

A197115 Nonsquare positive integers k such that the fundamental unit of the quadratic field Q(sqrt(k)) has norm -1 and can be written as x + y*sqrt(d) with integers x, y where d is the squarefree part of k.

Original entry on oeis.org

2, 8, 10, 17, 18, 26, 32, 37, 40, 41, 50, 58, 65, 68, 72, 73, 74, 82, 89, 90, 97, 98, 101, 104, 106, 113, 122, 128, 130, 137, 145, 148, 153, 160, 162, 164, 170, 185, 193, 197, 200, 202, 218, 226, 232, 233, 234, 241, 242, 250, 257, 260, 265, 269, 272, 274
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2011

Keywords

Comments

This sequence is a subsequence of A172000.

Crossrefs

Cf. A087643, A172000, A194366.

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, 400}]; cr

Extensions

Definition clarified by Emmanuel Vantieghem, Mar 06 2017

A197127 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))is singular.

Original entry on oeis.org

6, 14, 22, 30, 34, 38, 42, 46, 54, 56, 62, 66, 69, 70, 78, 86, 87, 93, 94, 102, 110, 114, 115, 118, 126, 130, 132, 134, 138, 142, 146, 150, 154, 155, 156, 158, 159, 166, 174, 177, 178, 182, 183, 184, 185, 186, 190, 194, 198, 206, 210, 214, 220, 222, 228, 230
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2011

Keywords

Comments

x^2+n*y^2=(+/-)2^s where s is 0 or 1.
Definition: Unity is singular when GCD[n,y]<>1.

Examples

			a(1)=6 because unity of quadratic field  Q(6) is 5+2*Sqrt[6] and GCD[2,6]=2 <>1.

Crossrefs

Cf. A087643, A172000, A194366, A197115, 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, n]]], {n, 2, 330}]; cr (*Artur Jasinski*)

A197128 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n))is not singular.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 63, 65, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 88, 89, 90, 91, 92
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2011

Keywords

Comments

x^2+n*y^2=(+/-)2^s where s is 0 or 1.
Definition: Unity is singular when GCD[n,y]<>1.

Crossrefs

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

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, n]]], {n, 2, 330}]; cr

A197170 Smallest k such that the fundamental unit (x+y*w) or (x+y*w)/2 of the real quadratic field Q(sqrt(k)) obeys gcd(k,y)=n.

Original entry on oeis.org

6, 69, 248, 115, 78, 511, 1016, 603, 70, 385, 3432, 793, 238, 2655, 14224, 1241, 3186, 703, 3980, 9177, 154, 736, 456, 1825, 3172, 13959, 2884, 319, 1110, 4619, 7136, 10659, 7174, 10255, 44856, 7067, 2926, 16185, 54280, 779, 7602, 10879, 22088, 10215, 46
Offset: 2

Views

Author

Artur Jasinski, Oct 11 2011

Keywords

Comments

Conjecture: For every n such a quadratic field with minimum k exists.

Examples

			For n=2 the unit is 2*w-5 with k=6.
For n=3 the unit is (3*w+25)/2 with k=69.
For n=4 the unit is (4*w-63) with k=248.
For n=5 the unit is 105*w-1126 with k=115.
For n=7 the unit is 185290497*w-4188548960 with k=511 (and this x and y appear in A041976 and A041977).

Crossrefs

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

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[AppendTo[aa, cr[[First[Position[ck, n]][[1]]]]], {n, 2, 99}]; aa

Formula

k = A197127(m) where m is the smallest m such that A197169(m)=n.

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

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
Showing 1-9 of 9 results.

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