A385354 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^2, where 0 < x < y < z has exactly n integer solutions.

6, 188, 768, 1728, 2640, 21120, 42336, 13824, 71280, 5832, 80352, 74088, 425088, 421875, 1058400, 110592, 287496, 46656

Offset: 1

Author

Zhining Yang, Jun 26 2025

Keywords

• nonn
• more

Comments

a(19) > 2000000, a(20) = 216000, a(22) = 884736.

Examples

			a(3)=768, because 768^2 = 54^3 + 59^3 + 61^3 = 40^3 + 62^3 + 66^3 = 24^3 + 40^3 + 80^3 and no integer less than 768 has 3 solutions.
		

Crossrefs

Cf. A024975, A025419, A377444.

Programs

Mathematica

s = Table[{k, Length@Select[PowersRepresentations[k^2, 3, 3],
     0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 2000}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 4}][[All, 1]]

Extensions

a(18) from Chai Wah Wu, Jul 05 2025