A385566 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^6, where 0 < x < y < z has exactly n integer solutions.

3, 6, 16, 12, 27, 63, 38, 24, 94, 18, 123, 42, 93, 75, 141, 48, 66, 36, 153, 60, 140, 96, 279, 114, 200, 138, 410, 174, 72, 126, 186, 168, 204, 150, 108, 426, 132, 220, 418, 246, 498, 736, 144, 120, 294, 306, 210, 666, 282, 378, 252, 770, 216, 460, 462, 534, 180

Offset: 1

Author

Zhining Yang, Jul 03 2025

Keywords

• nonn

Examples

			a(3)=16, because 16^6 = 9^3 + 58^3 + 255^3 = 9^3 + 183^3 + 220^3 = 22^3 + 57^3 + 255^3  and no integer less than 16 has 3 solutions.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..104

Crossrefs

Cf. A024975, A025419, A377444, A385354, A385565.

Programs

Mathematica

s = Table[{k, Length@Select[PowersRepresentations[k^6, 3, 3], 0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 30}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]

Extensions

a(41)-a(57) from Chai Wah Wu, Jul 07 2025