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 1-7 of 7 results.

A134042 Complete list of solutions to y^2 = x^3 + 113; sequence gives x values.

Original entry on oeis.org

-4, 2, 8, 11, 26, 422
Offset: 1

Views

Author

Artur Jasinski, Oct 03 2007

Keywords

Comments

For corresponding y values and examples see A134043.

Crossrefs

Cf. A134043, A029728, A080761.

Programs

  • Magma
    Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 113])) ]); /* adapted from A029728 */
  • SageMath
    [i[0] for i in EllipticCurve([0, 113]).integral_points()] # Seiichi Manyama, Aug 25 2019

Extensions

Edited and corrected by Klaus Brockhaus, Oct 04 2007

A080761 Positive numbers of the form y^2 - x^3, x and y >= 1.

Original entry on oeis.org

1, 3, 8, 9, 12, 15, 17, 18, 19, 22, 24, 28, 30, 35, 36, 37, 38, 40, 41, 44, 48, 54, 55, 56, 57, 63, 64, 65, 68, 71, 73, 79, 80, 89, 92, 94, 97, 98, 99, 100, 101, 105, 106, 107, 108, 112, 113, 117, 119, 120, 121, 128, 129, 131, 132, 136, 138, 141, 142, 143, 145, 148, 151
Offset: 1

Views

Author

Cino Hilliard, Mar 10 2003

Keywords

Comments

From Artur Jasinski, Oct 03 2007: (Start)
Some numbers have multiple partitions:
8 = 4^2 - 8^3 = 312^2 - 46^3,
9 = 6^2 - 3^3 = 15^2 - 6 ^3 = 253^2 - 40^3. (End)
This is Mordell's equation with the condition that x and y are positive. Sequence A054504 lists the n for which there is no solution to Mordell's equation. Hence, none of those numbers will be in this sequence. The terms of this sequence can be determined by looking at the link to Gebel's data. - T. D. Noe, Mar 23 2011

Examples

			8 is in the sequence since 3^2 = 1^3 + 8.

Links

Crossrefs

Complement of A080762.
Cf. sequences for n^3+7, n^3+17, n^3+3, n^3+2, n^3+5.
Cf. A029727, A029728, A134042, A134043.

Programs

  • Mathematica
    With[{nn=100},Take[Union[Select[First[#]^2-Last[#]^3&/@Tuples[Range[ 20nn],2],#>0&]],nn]] (* Harvey P. Dale, Jul 10 2012 *)
  • PARI
    diop(n,m) = { for(p=1,m, for(x=1,n, y=x*x*x+p; if(issquare(y),print1(p" "); break) ) ) }

Extensions

"Positive" added to definition by N. J. A. Sloane, Oct 06 2007

A134073 Complete list of solutions to y^2 = x^3 + 73; sequence gives y values.

Original entry on oeis.org

3, 9, 10, 17, 611, 6717
Offset: 1

Views

Author

Klaus Brockhaus, Oct 07 2007

Keywords

Comments

For corresponding x values see A134074.

Examples

			a(1)^2 = 3^2 = 9 = A134074(1)^3 + 73 = -64 + 73.
a(2)^2 = 9^2 = 81 = A134074(2)^3 + 73 = 8 + 73.
a(3)^2 = 10^2 = 100 = A134074(3)^3 + 73 = 27 + 73.
a(4)^2 = 17^2 = 289 = A134074(4)^3 + 73 = 216+ 73.
a(5)^2 = 611^2 = 373321 = A134074(5)^3 + 73 = 373248+ 73.
a(6)^2 = 6717^2 = 45118089 = A134074(6)^3 + 73 = 45118016+ 73.

Crossrefs

Cf. A134074, A029727, A134043.

Programs

  • Magma
    Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 73])) ]); /* adapted from A029727 */

A134106 Complete list of solutions to y^2 = x^3 - 207; sequence gives y values.

Original entry on oeis.org

3, 39, 75, 172, 5511, 6022, 223063347
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007

Keywords

Comments

For corresponding x values see A134107.

Examples

			a(1)^2 = 3^2 = 9 = A134107(1)^3 - 207 = 216 - 207.
a(2)^2 = 39^2 = 1521 = A134107(2)^3 - 207 = 1728 - 207.
a(3)^2 = 75^2 = 5625 = A134107(3)^3 - 207 = 5832 - 207.
a(4)^2 = 172^2 = 29584 = A134107(4)^3 - 207 = 29791 - 207.
a(5)^2 = 5511^2 = 30371121 = A134107(5)^3 - 207 = 30371328 - 207.
a(6)^2 = 6022^2 = 36264484 = A134107(6)^3 - 207 = 36264691 - 207.
a(7)^2 = 223063347^2 = 49757256774842409 = A134107(7)^3 - 207 = 49757256774842616 - 207.

Crossrefs

Cf. A134107, A134102, A134104, A029727, A134043, A134073.

Programs

  • Magma
    Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -207])) ]); /* adapted from A029727 */
  • Sage
    [x[1] for x in EllipticCurve([0,-207]).integral_points()] # Charles R Greathouse IV, Aug 09 2024

A134102 Complete list of solutions to y^2 = x^3 + 225; sequence gives y values.

Original entry on oeis.org

3, 10, 15, 17, 21, 35, 60, 165, 465, 2415, 6159, 6576, 611085363
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007

Keywords

Comments

For corresponding x values see A134103.

Examples

			a(1)^2 = 3^2 = 9 = A134103(1)^3 + 225 = -216 + 225.
a(2)^2 = 10^2 = 100 = A134103(2)^3 + 225 = -125 + 225.
a(3)^2 = 15^2 = 225 = A134103(3)^3 + 225 = 0 + 225.
a(4)^2 = 17^2 = 289 = A134103(4)^3 + 225 = 64 + 225.
a(5)^2 = 21^2 = 441 = A134103(5)^3 + 225 = 216 + 225.
a(6)^2 = 35^2 = 1225 = A134103(6)^3 + 225 = 1000 + 225.
a(7)^2 = 60^2 = 3600 = A134103(7)^3 + 225 = 3375 + 225.
a(8)^2 = 165^2 = 27225 = A134103(8)^3 + 225 = 27000 + 225.
a(9)^2 = 465^2 = 216225 = A134103(9)^3 + 225 = 216000 + 225.
a(10)^2 = 2415^2 = 5832225 = A134103(10)^3 + 225 = 5832000 + 225.
a(11)^2 = 6159^2 = 37933281 = A134103(11)^3 + 225 = 37933056 + 225.
a(12)^2 = 6576^2 = 43243776 = A134103(12)^3 + 225 = 43243551 + 225.
a(13)^2 = 611085363^2 = 373425320872841769 = A134103(13)^3 + 225 = 373425320872841544 + 225.

Crossrefs

Cf. A134103, A134104, A134106, A029727, A134043, A134073.

Programs

  • Magma
    { x : x in Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 225])) ]) }; /* adapted from A029727 */
  • Mathematica
    Select[Table[Sqrt[x^3+225],{x,-6,721000}],IntegerQ] (* Harvey P. Dale, Dec 25 2022 *)

A134104 Complete list of solutions to y^2 = x^3 + 297; sequence gives y values.

Original entry on oeis.org

9, 17, 18, 19, 45, 199, 333, 50265, 28748141
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007

Keywords

Comments

For corresponding x values see A134105.

Examples

			a(1)^2 = 9^2 = 81 = A134105(1)^3 + 297 = -216 + 297.
a(2)^2 = 17^2 = 289 = A134105(2)^3 + 297 = -8 + 297.
a(3)^2 = 18^2 = 324 = A134105(3)^3 + 297 = 27 + 297.
a(4)^2 = 19^2 = 361 = A134105(4)^3 + 297 = 64 + 297.
a(5)^2 = 45^2 = 2025 = A134105(5)^3 + 297 = 1728 + 297.
a(6)^2 = 199^2 = 39601 = A134105(6)^3 + 297 = 39304 + 297.
a(7)^2 = 333^2 = 110889 = A134105(7)^3 + 297 = 110592 + 297.
a(8)^2 = 50265^2 = 2526570225 = A134105(8)^3 + 297 = 2526569928 + 297.
a(9)^2 = 28748141^2 = 826455610955881 = A134105(9)^3 + 297 = 826455610955584 + 297.

Crossrefs

Cf. A134105, A134102, A134106, A029727, A134043, A134073.

Programs

  • Magma
    Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 297])) ]); /* adapted from A029727 */
  • Mathematica
    sol[x_] := Solve[y > 0 && x^3 - y^2 == -297, y, Integers];
Reap[For[x = 1, x < 10^5, x++, sx = sol[x]; If[sx != {}, xy = {x, y} /. sx[[1]]; Print[xy]; Sow[xy]]; sx = sol[-x]; If[sx != {}, xy = {-x, y} /. sx[[1]]; Print[xy]; Sow[xy]]]][[2, 1]][[All, 2]] // Sort (* Jean-François Alcover, Feb 07 2020 *)

A134166 Complete list of solutions to y^2 = x^3 + 1025; sequence gives y values.

Original entry on oeis.org

5, 30, 31, 32, 33, 45, 95, 255, 355, 513, 1930, 2139, 9419, 27905, 218796, 227805
Offset: 1

Views

Author

Klaus Brockhaus, Oct 11 2007

Keywords

Comments

For corresponding x values see A134167.

Examples

			a(1)^2 = 5^2 = 25 = A134167(1)^3 + 1025 = -1000 + 1025.
a(2)^2 = 30^2 = 900 = A134167(2)^3 + 1025 = -125 + 1025.
a(3)^2 = 31^2 = 961 = A134167(3)^3 + 1025 = -64 + 1025.
a(4)^2 = 32^2 = 1024 = A134167(4)^3 + 1025 = -1 + 1025.
a(5)^2 = 33^2 = 1089 = A134167(5)^3 + 1025 = 64 + 1025.
a(6)^2 = 45^2 = 2025 = A134167(6)^3 + 1025 = 1000 + 1025.
a(7)^2 = 95^2 = 9025 = A134167(7)^3 + 1025 = 8000 + 1025.
a(8)^2 = 255^2 = 65025 = A134167(8)^3 + 1025 = 64000 + 1025.
a(9)^2 = 355^2 = 126025 = A134167(9)^3 + 1025 = 125000 + 1025.
a(10)^2 = 513^2 = 263169 = A134167(10)^3 + 1025 = 262144 + 1025.
a(11)^2 = 1930^2 = 3724900 = A134167(11)^3 + 1025 = 3723875 + 1025.
a(12)^2 = 2139^2 = 4575321 = A134167(12)^3 + 1025 = 4574296 + 1025.
a(13)^2 = 9419^2 = 88717561 = A134167(13)^3 + 1025 = 88716536 + 1025.
a(14)^2 = 27905^2 = 778689025 = A134167(14)^3 + 1025 = 778688000 + 1025.
a(15)^2 = 218796^2 = 47871689616 = A134167(15)^3 + 1025 = 47871688591 + 1025.
a(16)^2 = 227805^2 = 51895118025 = A134167(16)^3 + 1025 = 51895117000 + 1025.

Crossrefs

Cf. A134167, A029727, A134043, A134073, A134102, A134104, A134106.

Programs

  • Magma
    { x : x in Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 1025])) ]) }; /* adapted from A029727 */
  • Mathematica
    Select[Table[Sqrt[1025+n^3],{n,-10,20000}],IntegerQ] (* Harvey P. Dale, Jan 21 2023 *)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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