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 1-5 of 5 results.

A134107 Complete list of solutions to y^2 = x^3 - 207; sequence gives x values.

Original entry on oeis.org

6, 12, 18, 31, 312, 331, 367806
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007

Keywords

Comments

For corresponding y values and examples see A134106.

Crossrefs

Cf. A134106, A134103, A134105, A029728, A134042, A134074.

Programs

  • Magma
    Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, -207])) ]); /* adapted from A029728 */
  • SageMath
    [i[0] for i in EllipticCurve([0, -207]).integral_points()] # Seiichi Manyama, Aug 25 2019

A134109 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 1, 0, 3, 2, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 3
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007, Oct 14 2007

Keywords

Comments

a(n) = A081120(n)/2 if A081120(n) is even, (A081120(n)+1)/2 if A081120(n) is odd (i.e. if n is a cubic number).
Comment from T. D. Noe, Oct 12 2007: In sequences A134108 and A134109 (this entry) dealing with the equation y^2 = x^3 + n, one could note that these are Mordell equations. Here are some related sequences: A054504, A081119, A081120, A081121. The link "Integer points on Mordell curves" has data on 20000 values of n. A134108 and A134109 count only solutions with y >= 0 and can be derived from A081119 and A081120.

Examples

			y^2 = x^3 - 4 has solutions (y, x) = (2, 2) and (11, 5), hence a(4) = 2.
y^2 = x^3 - 5 has no solutions, hence a(5) = 0.
y^2 = x^3 - 8 has solution (y, x) = (0, 2), hence a(8) = 1.
y^2 = x^3 - 207 has 7 solutions (see A134106, A134107), hence a(207) = 7.

Links

Crossrefs

Cf. A081120, A134106, A134107, A134108.

Programs

  • Magma
    [ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -n])) }: n in [1..104] ];
  • Mathematica
    A081120 = Cases[Import["https://oeis.org/A081120/b081120.txt", "Table"], {, }][[All, 2]];
a[n_] := With[{an = A081120[[n]]}, If[EvenQ[an], an/2, (an+1)/2]];
a /@ Range[10000] (* Jean-François Alcover, Nov 28 2019 *)

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 1-5 of 5 results.

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