%I #20 Sep 08 2022 08:45:32
%S 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,
%T 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,
%U 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
%N Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 + n.
%C 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).
%C 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.
%H Jean-François Alcover, <a href="/A134108/b134108.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>
%e y^2 = x^3 + 1 has solutions (y, x) = (0, -1), (1, 0) and (3, 2), hence a(1) = 3.
%e y^2 = x^3 + 6 has no solutions, hence a(6) = 0.
%e y^2 = x^3 + 17 has 8 solutions (see A029727, A029728), hence a(17) = 8.
%e y^2 = x^3 + 27 has solution (y, x) = (0, -3), hence a(27) = 1.
%t (* 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 *)
%t A081119 = Cases[Import["https://oeis.org/A081119/b081119.txt", "Table"], {_, _}][[All, 2]];
%t a[n_] := With[{an = A081119[[n]]}, If[EvenQ[an], an/2, (an + 1)/2]];
%t a /@ Range[10000] (* _Jean-François Alcover_, Nov 24 2019 *)
%o (Magma) [ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, n])) }: n in [1..122] ];
%Y Cf. A081119, A029727, A029728, A134109.
%K nonn
%O 1,1
%A _Klaus Brockhaus_, Oct 08 2007, Oct 14 2007