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 1-10 of 23 results. Next

A179162 a(n) = least positive k such that Mordell's equation y^2 = x^3 + k has exactly n integral solutions.

Original entry on oeis.org

6, 27, 2, 343, 12, 1, 37, 8, 24, 512, 9, 35611289, 73, 10218313, 315, 129554216, 17, 274625, 297, 17576000, 2817, 200201625, 1737
Offset: 0

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Comments

Additional known terms: a(24)=4481, a(26)=225, a(28)=2089, a(32)=1025.
For least positive k such that equation y^2 = x^3 - k has exactly n integral solutions, see A179175.
If n is odd, then a(n) is perfect cube. [Ray Chandler]

Links

Crossrefs

Cf. A054504, A081119, A179145-A179162.

Extensions

Edited and a(11), a(13), a(15), a(17), a(19), a(21) added by Ray Chandler, Jul 11 2010

A179163 Numbers k such that Mordell's equation y^2 = x^3 - k has exactly 1 integral solution.

Original entry on oeis.org

1, 8, 27, 64, 125, 512, 729, 1000, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 15625, 19683, 24389, 27000, 32768, 35937, 39304, 42875, 46656, 50653, 59319, 64000, 68921, 79507, 91125, 97336, 110592, 117649, 125000, 132651
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Comments

Contains all sixth powers: suppose that y^2 = x^3 - t^6, then (y/t^3)^2 = (x/t^2)^3 - 1. The elliptic curve Y^2 = X^3 - 1 has rank 0 and the only rational points on it are (1,0), so y^2 = x^3 - t^6 has only one solution (t^2,0). - Jianing Song, Aug 24 2022

Links

Crossrefs

Cf. A081120, A081121, A179163-A179175, A356713.
Complement of A179149 among the positive cubes.
Cf. also A179145, A356703.

Programs

  • Mathematica
    (* Assuming every term is a cube *) xmax = 2000; r[n_] := Reap[Do[rpos = Reduce[y^2 == x^3 - n, y, Integers]; If[rpos =!= False, Sow[rpos]]; rneg = Reduce[y^2 == (-x)^3 - n, y, Integers]; If[rneg =!= False, Sow[rneg]], {x, 1, xmax}]]; ok[1] = True; ok[n_] := Which[rn = r[n]; rn[[2]] === {}, False, Length[rn[[2]]] > 1, False, ! FreeQ[rn[[2, 1]], Or], False, True, True]; ok[n_ /; !IntegerQ[n^(1/3)]] = False; A179163 = Reap[Do[If[ok[n], Print[n]; Sow[n]], {n, 1, 140000}]][[2, 1]] (* Jean-François Alcover, Apr 12 2012 *)

Formula

a(n) = A356713(n)^3. - Jianing Song, Aug 24 2022

Extensions

Edited and extended by Ray Chandler, Jul 11 2010

A356709 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 1 integral solution.

Original entry on oeis.org

3, 5, 6, 12, 13, 15, 17, 19, 20, 24, 27, 29, 30, 31, 39, 41, 42, 43, 45, 47, 48, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 66, 67, 68, 69, 73, 75, 76, 77, 79, 80, 82, 83, 85, 87, 89, 93, 94, 96, 97, 101, 102, 103, 106, 107, 108, 109, 111, 113, 115, 116, 117, 118, 119
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Numbers k such that Mordell's equation y^2 = x^3 + k^3 has no solution other than the trivial solution (-k,0).
Cube root of A179145.

Examples

			3 is a term since the equation y^2 = x^3 + 3^3 has no solution other than (-3,0).

Links

Crossrefs

Cf. A081119, A179145, A179147, A179149, A179151, A356710, A356711, A356712.
Indices of 1 in A356706, of 0 in A356707, and of 1 in A356708.
Complement of A356720.
Cf. also A356713, A228948.

A179149 Numbers k such that Mordell's equation y^2 = x^3 + k has exactly 5 integral solutions.

Original entry on oeis.org

1, 64, 729, 1000, 2744, 4096, 15625, 21952, 35937, 46656, 50653, 64000, 117649, 262144, 343000, 531441, 592704, 681472, 729000, 753571, 1000000, 1124864, 1771561, 2000376, 2197000, 2299968, 2744000, 2985984, 3652264, 4096000, 4826809, 5451776, 6229504, 7189057, 7529536
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Comments

Contains all sixth powers: suppose that y^2 = x^3 + t^6, then (y/t^3)^2 = (x/t^2)^3 + 1. The elliptic curve Y^2 = X^3 + 1 has rank 0 and the only rational points on it are (-1,0), (0,+-1), and (2,+-3), so y^2 = x^3 + t^6 has 5 solutions (-t^2,0), (0,+-t^3), and (2*t^2,+-3*t^3). - Jianing Song, Aug 24 2022

Links

Crossrefs

Cf. A054504, A081119, A179145-A179162, A356711.

Formula

a(n) = A356711(n)^3.

Extensions

Edited and extended by Ray Chandler, Jul 11 2010
a(31)-a(35) from Max Alekseyev, Jun 01 2023

A356711 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 5 integral solutions.

Original entry on oeis.org

1, 4, 9, 10, 14, 16, 25, 28, 33, 36, 37, 40, 49, 64, 70, 81, 84, 88, 90, 91, 100, 104, 121, 126, 130, 132, 140, 144, 154, 160, 169, 176, 184, 193, 196
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Cube root of A179149.
Contains all squares: suppose that y^2 = x^3 + t^6, then (y/t^3)^2 = (x/t^2)^3 + 1. The elliptic curve Y^2 = X^3 + 1 has rank 0 and the only rational points on it are (-1,0), (0,+-1), and (2,+-3), so y^2 = x^3 + t^6 has 5 solutions (-t^2,0), (0,+-t^3), and (2*t^2,+-3*t^3).

Examples

			1 is a term since the equation y^2 = x^3 + 1^3 has 5 solutions (-1,0), (0,+-1), and (2,+-3).

Crossrefs

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356710, A356712.
Indices of 5 in A356706, of 2 in A356707, and of 3 in A356708.

Extensions

a(31)-a(35) from Max Alekseyev, Jun 01 2023

A356710 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 3 integral solutions.

Original entry on oeis.org

7, 11, 21, 22, 23, 26, 34, 35, 38, 44, 46, 63, 71, 74, 86, 92, 95, 99, 110, 122, 129, 136, 152, 155, 158, 170, 175, 177, 183, 189, 190, 198, 201, 203, 207, 211
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Cube root of A179147.

Examples

			7 is a term since the equation y^2 = x^3 + 7^3 has 3 solutions (-7,0) and (21,+-98).

Crossrefs

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356711, A356712.
Indices of 3 in A356706, of 1 in A356707, and of 2 in A356708.

Extensions

a(30)-a(36) from Max Alekseyev, Jun 01 2023

A356712 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 7 integral solutions.

Original entry on oeis.org

2, 18, 50, 56, 57, 98, 112, 114, 148, 162, 224, 228, 273, 280, 330, 336, 338, 364, 448, 504, 513, 578
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Cube root of A179151.

Examples

			2 is a term since the equation y^2 = x^3 + 2^3 has 3 solutions (-2,0), (1,+-3), (2,+-4), and (46,+-312).

Crossrefs

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356710, A356711.
Indices of 7 in A356706, of 3 in A356707, and of 4 in A356708.

A179147 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 3 integral solutions.

Original entry on oeis.org

343, 1331, 9261, 10648, 12167, 17576, 39304, 42875, 54872, 85184, 97336, 250047, 357911, 405224, 636056, 778688, 857375, 970299, 1331000, 1815848, 2146689, 2515456, 3511808, 3723875, 3944312, 4913000, 5359375, 5545233, 6128487, 6751269, 6859000, 7762392, 8120601, 8365427, 8869743, 9393931
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Links

Crossrefs

Cf. A054504, A081119, A179145-A179162.

Extensions

Edited and extended by Ray Chandler, Jul 11 2010
a(30)-a(36) from Max Alekseyev, Jun 01 2023

A179151 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 7 integral solutions.

Original entry on oeis.org

8, 5832, 125000, 175616, 185193, 941192, 1404928, 1481544, 3241792, 4251528, 11239424, 11852352, 20346417, 21952000, 35937000, 37933056, 38614472, 48228544, 89915392, 128024064, 135005697, 193100552
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Links

Crossrefs

Cf. A054504, A081119, A179145-A179162.

Extensions

Edited and a(3)-a(22) from Ray Chandler, Jul 11 2010

A356703 Numbers k such that Mordell elliptic curve y^2 = x^3 + k has a number of integral points that is both odd and > 1.

Original entry on oeis.org

1, 8, 64, 343, 512, 729, 1000, 1331, 2744, 4096, 5832, 9261, 10648, 12167, 15625, 17576, 21952, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 64000, 85184, 97336, 117649, 125000, 175616, 185193, 250047, 262144, 274625, 343000, 357911, 373248, 405224, 474552, 531441, 592704, 636056
Offset: 1

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

Cubes k such that y^2 = x^3 + k has a solution other than (-k^(1/3), 0).
Contains all sixth powers since A179149 does.

Examples

			512 is a term since the equation y^2 = x^3 + 512 has 9 integral solutions (-8,0), (-7,+-13), (4,+-24), (8,+-32), and (184,+-2496).

Links

Crossrefs

Complement of A179145 among the positive cubes.
Cf. A081119, A179147, A179149, A179151, A179163, A179419.
Cf. also A356709, A356720, A356713, A228948.

Formula

a(n) = A356720(n)^3.
Showing 1-10 of 23 results. Next

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