A385565 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^4, where 0 < x < y < z has exactly n integer solutions.

11, 21, 64, 144, 330, 846, 342, 252, 1331, 1008, 720, 1890, 3780, 729, 4200, 2016, 1000, 216, 6300, 8352, 10800, 12312, 8568, 19440, 8280, 9576, 21204

Offset: 1

Author

Zhining Yang, Jul 03 2025

Keywords

• nonn
• more

Comments

a(13) and a(15) not found up to k = 3300, a(14) = 729, a(16) = 2016, a(17) = 1000, a(18) = 216.

Examples

			a(3)=64, because 64^4 = 9^3 + 58^3 + 255^3 = 9^3 + 183^3 + 220^3 = 22^3 + 57^3 + 255^3 and no integer less than 64 has 3 solutions.
		

Crossrefs

Cf. A024975, A025419, A377444, A385354, A385566.

Programs

Mathematica

s = Table[{k, Length@Select[PowersRepresentations[k^4, 3, 3],
      0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 400}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]

Extensions

a(13), a(15), a(19)-a(21) from Chai Wah Wu, Jul 08 2025

a(22)-a(27) from Chai Wah Wu, Jul 18 2025