A081120 -id:A081120 - cp's OEIS frontend

A356702 Records values in A081120.

1, 2, 4, 6, 8, 14, 18, 20, 22

Offset: 1

Jianing Song, Aug 23 2022

			a(9) = 22 since A356700(9) = 3807, and the equation y^2 = x^3 - 3807 has 22 integral solutions.

Cf. A081119, A081120, A356699, A356700, A356701.

A081119 Number of integral solutions to Mordell's equation y^2 = x^3 + n.

Original entry on oeis.org

5, 2, 2, 2, 2, 0, 0, 7, 10, 2, 0, 4, 0, 0, 4, 2, 16, 2, 2, 0, 0, 2, 0, 8, 2, 2, 1, 4, 0, 2, 2, 0, 2, 0, 2, 8, 6, 2, 0, 2, 2, 0, 2, 4, 0, 0, 0, 2, 2, 2, 0, 2, 0, 2, 2, 2, 6, 0, 0, 0, 0, 0, 4, 5, 8, 0, 0, 4, 0, 0, 2, 2, 12, 0, 0, 2, 0, 0, 2, 8, 2, 2, 0, 0, 0, 0, 0, 0, 8, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 12

Offset: 1

T. D. Noe, Mar 06 2003

Mordell's equation has a finite number of integral solutions for all nonzero n.
Gebel, Petho, and Zimmer (1998) computed the solutions for |n| <= 10^4. Bennett and Ghadermarzi (2015) extended this bound to |n| <= 10^7.
Sequence A054504 gives n for which there are no integral solutions. See A081120 for the number of integral solutions to y^2 = x^3 - n.
a(n) is odd iff n is a cube. - Bernard Schott, Nov 23 2019
From Jianing Song, Aug 24 2022: (Start)
a(n) = 5 if n is a sixth power. Further more, if A060950(n) = 0 (namely the elliptic curve y^2 = x^3 + n has rank 0), then:
- a(n) = 2 if n is a square but not a sixth power;
- a(n) = 1 if n is a cube but not a sixth power;
- a(n) = 0 otherwise.
This follows from the complete description of the torsion group of y^2 = x^3 + n, using O to denote the point at infinity (see Exercise 10.19 of Chapter X of Silverman's Arithmetic of elliptic curves):
- If n = t^6 is a sixth power, then the torsion group consists of O, (2*t^2,+-3*t^3), (0,+-t^3), and (-t^2, 0).
- If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).
- If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).
- If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).
- In all the other cases, the torsion group is trivial.
So a torsion point on y^2 = x^3 + n other than O is an integral point. If y^2 = x^3 + n has rank 0, then all the integral points on y^2 = x^3 + n are exactly the torsion points other than O.
Note that this result implies particularly that a(n) = a(n*t^6) for all t if A060950(n) = 0: the elliptic curve y^2 = x^3 + n*t^6 can be written as (y/t^3)^2 = (x/t^2)^3 + n, so it has the same Mordell-Weil group (hence the same rank and isomorphic torsion group) as y^2 = x^3 + n. (End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367.

Jean-François Alcover, Table of n, a(n) for n = 1..10000 [There were errors in the previous b-file, which had 10000 terms contributed by T. D. Noe and was based on the work of J. Gebel.]
Michael A. Bennett, Amir Ghadermarzi Mordell's equation: a classical approach. LMS J. Compute. Math. 18 (2015): 633-646. doi:10.1112/S1461157015000182 arXiv:1311.7077
J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
J. Gebel, A. Pethö and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367. (doi:10.1023/A:1000281602647 not working as of July 2024.)
Joseph H. Silverman, The Arithmetic of Elliptic Curves.
Eric Weisstein's World of Mathematics, Mordell Curve.
Wikipedia, Mordell curve.

Cf. A054504, A081120. See A134108 for another version.

(* This naive approach gives correct results up to n = 1000 *) xmax[] = 10^4; Do[xmax[n] = 10^5, {n, {297, 377, 427, 885, 899}}]; Do[xmax[n] = 10^6, {n, {225, 353, 618 }}]; f[n] := (x = -Ceiling[n^(1/3)]-1; s = {}; While[x <= xmax[n], x++; y2 = x^3 + n; If[y2 >= 0, y = Sqrt[y2]; If[ IntegerQ[y], AppendTo[s, y]]]]; s); a[n_] := (fn = f[n];  If[fn == {}, 0, 2 Length[fn] - If[First[fn] == 0, 1, 0] ]); Table[an = a[n]; Print["a[", n, "] = ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Oct 18 2011 *)

Edited by Max Alekseyev, Feb 06 2021

A081121 Numbers k such that Mordell's equation y^2 = x^3 - k has no integral solutions.

Original entry on oeis.org

3, 5, 6, 9, 10, 12, 14, 16, 17, 21, 22, 24, 29, 30, 31, 32, 33, 34, 36, 37, 38, 41, 42, 43, 46, 50, 51, 52, 57, 58, 59, 62, 65, 66, 68, 69, 70, 73, 75, 77, 78, 80, 82, 84, 85, 86, 88, 90, 91, 92, 93, 94, 96, 97, 98, 99

Offset: 1

T. D. Noe, Mar 06 2003

Mordell's equation has a finite number of integral solutions for all nonzero k. Gebel computes the solutions for k < 10^5. Sequence A054504 gives k for which there are no integral solutions to y^2 = x^3 + k. See A081120 for the number of integral solutions to y^2 = x^3 - n.
This is the complement of A106265. - M. F. Hasler, Oct 05 2013
Numbers k such that A081120(k) = 0. - Charles R Greathouse IV, Apr 29 2015

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.

T. D. Noe, Table of n, a(n) for n = 1..7757 (from Gebel, 3136 and 6789 removed by _Seth A. Troisi_, May 20 2019)
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
J. Gebel, A. Petho and G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367. MR1602064.
Eric Weisstein's World of Mathematics, Mordell Curve

Cf. A054504, A081120, A106265.

m = 99; f[_List] := (xm = 2 xm; ym = Ceiling[xm^(3/2)];
Complement[Range[m], Outer[Plus, -Range[0, ym]^2, Range[-xm, xm]^3] //Flatten //Union]); xm=10; FixedPoint[f, {}] (* Jean-François Alcover, Apr 29 2011 *)

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

Artur Jasinski, Jun 30 2010

nonn

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

Jianing Song, Table of n, a(n) for n = 1..108 (using data from A179149)
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

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

(* 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 *)

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

Edited and extended by Ray Chandler, Jul 11 2010

A179175 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

3, 1, 2, 1331, 4, 216, 28, 54872, 116, 343, 828, 250047, 496, 71991296, 207

Offset: 0

Artur Jasinski, Jun 30 2010

The status of further terms is:
  15 integral solutions: unknown
  16 integral solutions: 503
  17 integral solutions: unknown
  18 integral solutions: 431
  19 integral solutions: unknown
  20 integral solutions: 2351
  21 integral solutions: unknown
  22 integral solutions: 3807
For least positive k such that equation y^2 = x^3 + k has exactly n integral solutions, see A179162.
If n is odd, then a(n) is perfect cube. [Ray Chandler]
From Jose Aranda, Aug 04 2024: (Start)
About those unknown terms:
a(15) <=    2600^3 = (26* 10^2)^3
a(17) <=   10400^3 = (26* 20^2)^3
a(19) <=   93600^3 = (26* 60^2)^3
a(21) <= 4586400^3 = (26*420^2)^3
The term a(13) = 71991296 = 416^3 = (26*4^2)^3. (End)

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A081120, A081121, A179163-A179174.

Edited and a(7), a(11), a(13) added by Ray Chandler, Jul 11 2010

A060951 Rank of elliptic curve y^2 = x^3 - n.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2

Offset: 1

N. J. A. Sloane, May 10 2001

The curves for n and -27*n are isogenous (as Noam Elkies points out--see Womack), so they have the same rank. - Jonathan Sondow, Sep 10 2013

			a(1) = A060950(27) = a(729) = 0. - _Jonathan Sondow_, Sep 10 2013

T. D. Noe, Table of n, a(n) for n = 1..10000 (from Gebel)
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
H. Mishima, Tables of Elliptic Curves
T. Womack, Minimal-known positive and negative k for Mordell curves of given rank

Cf. A031508, A002150, A002152, A002154, A179136, A179137.
Cf. A060748, A060838, A060950, A060952, A060953.
Cf. A081120 (number of integral solutions to Mordell's equation y^2 = x^3 - n).

{a(n) = if( n<1, 0, length( ellgenerators( ellinit( [ 0, 0, 0, 0, -n], 1))))} /* Michael Somos, Mar 17 2011 */

apply( {A060951(n)=ellrank(ellinit([0,-n]))[1]}, [1..99]) \\ For version < 2.14, use ellanalyticrank(...). - M. F. Hasler, Jul 01 2024

a(n) = A060950(27*n) and A060950(n) = a(27*n), so a(n) = a(729*n). - Jonathan Sondow, Sep 10 2013

Corrected Apr 08 2005 at the suggestion of James R. Buddenhagen. There were errors caused by the fact that Mishima lists each curve of rank two twice, once for each generator.

A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 15, 18, 19, 20, 23, 25, 26, 27, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 64, 67, 71, 72, 74, 76, 79, 81, 83, 87, 89, 95, 100, 104, 106, 107, 109, 112, 116, 118, 121, 124, 125, 126, 127, 128, 135, 139, 143, 146, 147, 148, 150, 151, 152, 153

Offset: 1

Zak Seidov, Apr 28 2005

nonn

A given a(n) can have multiple solutions with distinct (b,c), e.g., a=4 with b=2, c=2 (4 + 2^2 = 2^3) or with b=11, c=5 (4 + 11^2 = 5^3). (See also A181138.) Sequences A106266 and A106267 list the minimal values. - M. F. Hasler, Oct 04 2013
The cubes A000578 = (1, 8, 27, 64, ...) form a subsequence of this sequence, corresponding to b=0, a=c^3. If b=0 is excluded, these terms are not present, except for a few exceptions, a = 216, 343, 12167, ... (6^3 + 28^2 = 10^3, 7^3 + 13^2 = 8^3, 23^3 + 588^2 = 71^3, ...), cf. A038597 for the possible b-values. - M. F. Hasler, Oct 05 2013
This is the complement of A081121. The values do indeed correspond to solutions listed in Gebel's file. - M. F. Hasler, Oct 05 2013
B-file corrected following a remark by Alois P. Heinz, May 24 2019. A double-check would be appreciated in view of two values that were missing, for unknown reasons, in the earlier version of the b-file. - M. F. Hasler, Aug 10 2024

			a = 1,2,4,7,8,11,13,15,18,19,20,23,25,26,27,28,35,39,40,44,45,47,48,49,53, ...
b = 0,5,2,1,0, 4,70, 7, 3,18,14, 2,10, 1, 0, 6,36, 5,52, 9,96,13,4,524,26, ...
c = 1,3,2,2,2, 3,17, 4, 3, 7, 6, 3, 5, 3, 3, 4,11, 4,14, 5,21, 6, 4,65, 9, ...
Here are the values grouped together:
{{1, 0, 1}, {2, 5, 3}, {4, 2, 2}, {7, 1, 2}, {8, 0, 2}, {11, 4, 3}, {13, 70, 17}, {15, 7, 4}, {18, 3, 3}, {19, 18, 7}, {20, 14, 6}, {23, 2, 3}, {25, 10, 5}, {26, 1, 3}, {27, 0, 3}, {28, 6, 4}, {35, 36, 11}, {39, 5, 4}, {40, 52, 14}, {44, 9, 5}, {45, 96, 21}, {47, 13, 6}, {48, 4, 4}, {49, 524, 65}, {53, 26, 9}, {54, 17, 7}, {55, 3, 4}, {56, 76, 18}, {60, 2, 4}, {61, 8, 5}, {63, 1, 4}, {64, 0, 4}, {67, 110, 23}, {71, 21, 8}, ... }
a(2243) = 10000 = 25^3 - 75^2. - _M. F. Hasler_, Oct 05 2013, index corrected Aug 10 2024
a(136) = 366 = 11815^3 - 1284253^2 (has c/a(n) ~ 32.3); a(939) = 3607 = 244772^3 - 121099571^2 (has c/a(n) ~ 67.9); a(1090) = 4265 = 84521^3 - 24572364^2 (has c/a(n) ~ 19.8). - _M. F. Hasler_, Aug 10 2024

M. F. Hasler, Table of n, a(n) for n = 1..2500 (corrected and extended Aug 10 2024)
J. Gebel, Integer points on Mordell curves, negative k values [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A106266, A106267 for respective minimal values of b and c.
Cf. A023055: (Apparent) differences between adjacent perfect powers (integers of form a^b, a >= 1, b >= 2); A076438: n which appear to have a unique representation as the difference of two perfect powers; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1; A076440: n which appear to have a unique representation as the difference of two perfect powers and one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1 and that solution has odd x or odd y (or both odd); A075772: Difference between n-th perfect power and the closest perfect power, etc.
Cf. A023055, A075772, A076438, A076440.
Cf. A054504, A081121, A081120; A179386 - A179388.

f[n_] := Block[{k = Floor[n^(1/3) + 1]}, While[k < 10^6 && !IntegerQ[ Sqrt[k^3 - n]], k++ ]; If[k == 10^6, 0, k]]; Select[ Range[ 154], f[ # ] != 0 &] (* Robert G. Wilson v, Apr 28 2005 *)

select( {is_A106265(a, L=99)=for(c=sqrtnint(a, 3), (a+9)*L, issquare(c^3-a, &b) && return(c))}, [1..199]) \\ The function is_A106265 returns 0 if n isn't a term, or else the c-value (A106267) which can't be zero if n is a term. The L-value can be used to increase the search limit but so far no instance is known that requires L>68. - M. F. Hasler, Aug 10 2024

a(n) = A106267(n)^3 - A106266(n)^2.

More terms from Robert G. Wilson v, Apr 28 2005

Definition corrected, solutions with b=0 added by M. F. Hasler, Sep 30 2013

A134108 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 + n.

Original entry on oeis.org

3, 1, 1, 1, 1, 0, 0, 4, 5, 1, 0, 2, 0, 0, 2, 1, 8, 1, 1, 0, 0, 1, 0, 4, 1, 1, 1, 2, 0, 1, 1, 0, 1, 0, 1, 4, 3, 1, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 3, 0, 0, 0, 0, 0, 2, 3, 4, 0, 0, 2, 0, 0, 1, 1, 6, 0, 0, 1, 0, 0, 1, 4, 1, 1, 0, 0, 0, 0, 0, 0, 4, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 6, 2, 0, 0, 0, 1

Offset: 1

Klaus Brockhaus, Oct 08 2007, Oct 14 2007

nonn

a(n) = A081119(n)/2 if A081119(n) is even, (A081119(n)+1)/2 if A081119(n) is odd (i.e. if n is a cubic number).

Comment from T. D. Noe, Oct 12 2007: In sequences A134108 (this entry) and A134109 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.

			y^2 = x^3 + 1 has solutions (y, x) = (0, -1), (1, 0) and (3, 2), hence a(1) = 3.
y^2 = x^3 + 6 has no solutions, hence a(6) = 0.
y^2 = x^3 + 17 has 8 solutions (see A029727, A029728), hence a(17) = 8.
y^2 = x^3 + 27 has solution (y, x) = (0, -3), hence a(27) = 1.

Jean-François Alcover, Table of n, a(n) for n = 1..10000
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Eric Weisstein's World of Mathematics, Mordell Curve

Cf. A081119, A029727, A029728, A134109.

[ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, n])) }: n in [1..122] ];

(* This naive approach gives correct results up to n=1000 *) xmax[] = 10^4; Do[ xmax[n] = 10^5, {n, {297, 377, 427, 885, 899}}]; Do[ xmax[n] = 10^6, {n, {225, 353, 618}}]; f[n] := (x = -Ceiling[n^(1/3)] - 1; s = {}; While[x <= xmax[n], x++; y2 = x^3 + n; If[y2 >= 0, y = Sqrt[y2]; If[IntegerQ[y], AppendTo[s, y]]]]; s); a[n_] := a[n] = (fn = f[n]; an = If[fn == {}, 0, 2 Length[fn] - If[First[fn] == 0, 1, 0]]; If[EvenQ[an], an/2, (an + 1)/2]); Table[ Print["a[", n, "] = ", a[n] ]; a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 20 2012 *)
A081119 = Cases[Import["https://oeis.org/A081119/b081119.txt", "Table"], {, }][[All, 2]];
a[n_] := With[{an = A081119[[n]]}, If[EvenQ[an], an/2, (an + 1)/2]];
a /@ Range[10000] (* Jean-François Alcover, Nov 24 2019 *)

A229618 Numbers that are the distance between a square and the next larger cube.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 15, 18, 19, 20, 25, 26, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 67, 71, 72, 74, 76, 79, 81, 83, 87, 100, 104, 106, 107, 109, 112, 116, 118, 126, 127, 128, 135, 139, 143

Offset: 1

M. F. Hasler, Sep 26 2013

This is the range of the sequence A181138 (= least k>0 such that n^2+k is a cube). Note that this is not the same as A087285 = range of A077116 = difference between a cube and the next smaller square: If n^2+k = y^3 is the smallest cube above n^2, then n^2 is not necessarily the largest square below y^3, e.g., 9+18 = 27 = 3^3 is the least cube above 9 = 3^2, but 25 = 5^2 is the largest square below 27. Therefore the number 18 is in this sequence, but not in A087285.
See A077116 and A181138 and A179386 for motivations.
Apart from the leading 1, this is a subsequence of A106265, which does not require the square to be the next smaller one: For example, 23 = 27 - 4 = 3^3 - 2^2 is in A106265 but not in this sequence. A165288 is a subsequence of this one, except for the initial term.

			a(1) = 1 = 1^3-0^2 (but this is the only solution to y^3-x^2 = 1).
a(2) = 2 = 27-25 (= 3^3-5^2), and this is the only solution to y^3-x^2 = 2.
The number 3 is not in the sequence since there are no x, y > 0 such that y^3-x^2 = 3.
a(3) = 4 = 8-4 (= 2^3-2^2) = 125-121 (= 5^3-11^2); these are the only two solutions to y^3-x^2 = 4, for all x>11, the minimal positive y^3-x^2 is 7.

Cf. A087285, A087286, A088017, A081121, A081120, A077116, A065733.

A356713 Numbers k such that Mordell's equation y^2 = x^3 - k^3 has exactly 1 integral solution.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 1

Jianing Song, Aug 23 2022

nonn

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 A179163.
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), so y^2 = x^3 - t^6 has only one solution (t^2,0).

Jianing Song, Table of n, a(n) for n = 1..108 (using data from A179149)

Cf. A081120, A179163, A356709, A356720. Complement of A228948.

1 is a term since the equation y^2 = x^3 - 1^3 has no solution other than (1,0).

cp's OEIS Frontend

A356702 Records values in A081120.

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A081119 Number of integral solutions to Mordell's equation y^2 = x^3 + n.

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A081121 Numbers k such that Mordell's equation y^2 = x^3 - k has no integral solutions.

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

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A179175 a(n) = least positive k such that Mordell's equation y^2 = x^3 - k has exactly n integral solutions.

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A060951 Rank of elliptic curve y^2 = x^3 - n.

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A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

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Mathematica

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A134108 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 + n.

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