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 14 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

A179174 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 22 integral solutions.

Original entry on oeis.org

3807, 3896, 52784, 129556, 157239, 167600, 185112, 200871, 281439, 314199, 347967, 370647, 399375, 553648, 623872, 720703, 815728, 819775, 856799, 934975, 994816
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Comments

Counting (+x,+y) and (+x,-y) iff y != 0.

Links

Crossrefs

Subsequence of A007412.
Cf. A081120, A081121, A179163-A179175, A134107, A106265.

Extensions

Edited by Ray Chandler, Jul 11 2010
a(3)-a(21) from Jose Aranda, Aug 10 2024

A179164 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 2 integral solutions.

Original entry on oeis.org

2, 13, 15, 18, 19, 20, 23, 25, 35, 40, 44, 45, 49, 54, 56, 61, 67, 71, 72, 74, 79, 81, 83, 87, 89, 95, 106, 107, 112, 118, 121, 124, 126, 127, 128, 139, 143, 146, 148, 150, 151, 153, 155, 159, 167, 170, 172, 175, 184, 186, 188, 193, 199, 222, 223, 233, 235, 236, 239
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Links

Crossrefs

Cf. A081120, A081121, A179163-A179175.

Extensions

Edited by Ray Chandler, Jul 11 2010

A356704 a(n) is the least k such that Mordell's equation y^2 = x^3 + k^3 has exactly 2*n+1 integral solutions.

Original entry on oeis.org

3, 7, 1, 2, 8, 329, 217, 506, 65, 260, 585
Offset: 0

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

a(n) is the least k such that y^2 = x^3 + k^3 has exactly n solutions with y positive, or exactly n+1 solutions with y nonnegative.
a(n) is the smallest index of 2*n+1 in A356706, of n in A356707, and of n+1 in A356708.

Examples

			a(4) = 8 since y^2 = x^3 + 8^3 has exactly 9 solutions (-8,0), (-7,+-13), (4,+-24), (8,+-32), and (184,+-2496), and the number of solutions to y^2 = x^3 + k^3 is not 9 for 0 < k < 8.

Crossrefs

Cf. A081119, A081120, A179162, A179175, A356705, A356706, A356707, A356708.

Formula

a(n) = A179162(2*n+1)^(1/3).

A356705 a(n) is the least k such that Mordell's equation y^2 = x^3 - k^3 has exactly 2*n+1 integral solutions.

Original entry on oeis.org

1, 11, 6, 38, 7, 63, 416, 2600, 10400, 93600
Offset: 0

Views

Author

Jianing Song, Aug 23 2022

Keywords

Comments

a(n) is the least k such that y^2 = x^3 - k^3 has exactly n solutions with y positive, or exactly n+1 solutions with y nonnegative.

Examples

			a(1) = 11 since y^2 = x^3 - 11^3 has exactly 3 solutions (11,0) and (443,+-9324), and the number of solutions to y^2 = x^3 - k^3 is not 3 for 0 < k < 11.
a(2) = 6 since y^2 = x^3 - 6^3 has exactly 5 solutions (6,0), (10,+-28), and (33,+-189), and the number of solutions to y^2 = x^3 - k^3 is not 5 for 0 < k < 6.
a(4) = 7 since y^2 = x^3 - 7^3 has exactly 9 solutions (7,0), (8,+-13), (14,+-49), (28,+-147), and (154,+-1911), and the number of solutions to y^2 = x^3 - k^3 is not 9 for 0 < k < 7.

Crossrefs

Cf. A081119, A081120, A179162, A179175, A356704.

Formula

a(n) = A179175(2*n+1)^(1/3).

Extensions

a(7)-a(9) from Jose Aranda, Aug 05 2024

A179165 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 4 integral solutions.

Original entry on oeis.org

4, 7, 11, 26, 48, 53, 55, 60, 63, 76, 109, 147, 180, 212, 215, 242, 256, 277, 362, 364, 375, 391, 405, 433, 448, 471, 476, 511, 535, 593, 615, 674, 680, 704, 728, 767, 782, 802, 831, 856, 875, 895, 900, 914, 931, 975, 991, 996, 1055, 1096, 1108, 1144, 1152
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Links

Crossrefs

Cf. A081120, A081121, A179163-A179175.

Extensions

Edited by Ray Chandler, Jul 11 2010

A179166 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 6 integral solutions.

Original entry on oeis.org

28, 39, 47, 100, 104, 135, 152, 174, 191, 200, 244, 424, 440, 459, 732, 755, 804, 888, 984, 1048, 1075, 1084, 1236, 1259, 1287, 1322, 1432, 1503, 1668, 1763, 1792, 1812, 1951, 2160, 2224, 2344, 2367, 2440, 2468, 2496, 2556, 2692, 2695, 2699, 2727, 2799
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Links

Crossrefs

Cf. A081120, A081121, A179163-A179175.

Extensions

Edited by Ray Chandler, Jul 11 2010

A179168 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 8 integral solutions.

Original entry on oeis.org

116, 307, 508, 516, 639, 964, 1192, 1208, 1588, 1607, 1692, 1719, 2036, 2548, 2575, 2708, 2888, 3231, 3376, 3623, 4598, 4743, 5319, 5508, 5823, 5872, 5887, 6012, 6400, 6551, 6691, 6823, 6911, 7375, 7600, 7740, 7900, 7975, 9127, 9408, 9455
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Links

Crossrefs

Cf. A081120, A081121, A179163-A179175.

Extensions

Edited by Ray Chandler, Jul 11 2010

A179169 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 10 integral solutions.

Original entry on oeis.org

828, 944, 980, 1724, 2188, 3051, 3471, 3952, 3967, 4031, 4080, 5095, 5296, 5975, 6908, 7100, 7424, 7516, 7775, 8623
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Links

Crossrefs

Cf. A081120, A081121, A179163-A179175.

Extensions

Edited by Ray Chandler, Jul 11 2010
Showing 1-10 of 14 results. Next

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