A392549 a(n) is the least positive integer k such that Mordell's equation y^2 = x^3 + k^2 has exactly n integer solutions with y >= 0.

2, 11, 1, 6, 3, 10, 80, 62, 63, 210, 55, 840, 15, 440, 120, 960, 3240, 561, 2415, 510, 665, 19320, 1155, 5320, 148785, 31185, 9240, 665000, 73920, 143640, 249480, 379610, 54285

Offset

1,1

Comments

a(34) > 10^6 if it exists, a(37) = 255255, a(38) = 434280, a(42) = 794310, a(64) = 451605.

Links

Table of n, a(n) for n=1..33.

Example

a(6) = 10 because 10 is the least k such that y^2 = x^3 + k^2 has 6 integral solutions with nonnegative y: {{-4, 6}, {0, 10}, {5, 15}, {20, 90}, {24, 118}, {2660, 137190}}.

Mathematica

f[k_]:=(m=Range[0, 71];
xmax=50000; (* This xmax only gives correct results up to n = 12, as n=13 has a larger solution {720114, 611085363} requiring a larger xmax *)
m72k=Intersection[Union[Mod[m^2-k^2, 72]], Union[Mod[m^3, 72]]];
s=Select[Range[-Floor[k^(2/3)], xmax], MemberQ[m72k, Mod[#^3, 72]]&]; t=Select[s, IntegerQ[Sqrt[#^3+k^2]]&]; {k, Length@t}); v=Table[f[k], {k, 100}];
a=Table[SelectFirst[v, #[[2]]==n&], {n, 9}][[All, 1]]

Crossrefs

Cf. A081119, A134108, A392548.

Sequence in context: A326042 A350035 A353877 * A250221 A365610 A298444

Adjacent sequences: A392546 A392547 A392548 * A392550 A392551 A392552

Keyword

nonn,more

Author

Zhining Yang, Jan 15 2026

Status

approved