A383877 a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^3, where 0 < x < y < z < w has exactly n integer solutions, or 0 if there is no such k.

14, 13, 55, 26, 52, 63, 70, 66, 56, 104, 102, 143, 161, 91, 117, 112, 78, 236, 180, 217, 198, 192, 140, 292, 216, 259, 156, 196, 344, 168, 210, 264, 325, 252, 406, 360, 380, 402, 315, 338, 234, 308, 351, 182, 396, 408, 399, 432, 441, 312, 474, 636, 513, 273, 336, 476, 618, 666

Offset: 1

Author

Zhining Yang, May 13 2025

Keywords

• nonn

Comments

The largest term for k<=10000 is a(3569)=9828.

Conjecture: a(n) != 0 for all n.

Examples

			a(3)=55, because 55^3 = 7^3 + 24^3 + 38^3 + 46^3 = 7^3 + 12^3 + 34^3 + 50^3 = 17^3 + 19^3 + 28^3 + 51^3 and no integer less than 55 has 3 solutions.
		

Links

• Zhining Yang, Table of n, a(n) for n = 1..1280

Crossrefs

Cf. A377444, A384439.

Programs

Mathematica

s=Table[{k,Length@Select[PowersRepresentations[k^3,4,3],0<#[[1]]<#[[2]]<#[[3]]<#[[4]]&]},{k,100}];
a=Table[SelectFirst[s,#[[2]]==k&],{k,9}][[All,1]]