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.
Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 73])) ]); /* adapted from A029728 */
[i[0] for i in EllipticCurve([0, 73]).integral_points()] # Seiichi Manyama, Aug 25 2019
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.
Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -207])) ]); /* adapted from A029727 */
[x[1] for x in EllipticCurve([0,-207]).integral_points()] # Charles R Greathouse IV, Aug 09 2024
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.
{ x : x in Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 225])) ]) }; /* adapted from A029727 */
Select[Table[Sqrt[x^3+225],{x,-6,721000}],IntegerQ] (* Harvey P. Dale, Dec 25 2022 *)
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.
Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 297])) ]); /* adapted from A029727 */
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 *)
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.
{ x : x in Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 1025])) ]) }; /* adapted from A029727 */
Select[Table[Sqrt[1025+n^3],{n,-10,20000}],IntegerQ] (* Harvey P. Dale, Jan 21 2023 *)
Comments