A385409 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^2, where 0 < x < y < z < w has exactly n integer solutions.

10, 42, 39, 153, 126, 276, 273, 312, 315, 476, 588, 336, 546, 777, 1053, 756, 1216, 1386, 1560, 1134, 1323, 1488, 1365, 1368, 1344, 1596, 2366, 2496, 2988, 1680, 2548, 1736, 2184, 3003, 3720, 2520, 3185, 3552, 2268, 3564, 4095, 3213, 4578, 4392, 5208, 4004, 4599, 5733

Offset: 1

Author

Zhining Yang, Jun 27 2025

Keywords

• nonn

Comments

Conjecture: a(n) exists for all n.

Examples

			a(4)=153, because 153^2 = 5^3 + 15^3 + 21^3 + 22^3 = 2^3 + 7^3 + 15^3 + 27^3 = 6^3 + 8^3 + 9^3 + 28^3 = 1^3 + 5^3 + 11^3 + 28^3 and no integer less than 153 has 4 solutions.
		

Links

• Zhining Yang, Table of n, a(n) for n = 1..1341

Crossrefs

Cf. A024975, A025419, A377444, A384430, A385354.

Programs

Mathematica

s = Table[{k, Length@Select[PowersRepresentations[k^2, 4, 3],
      0 < #[[1]] < #[[2]] < #[[3]] < #[[4]] &]}, {k, 500}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 10}][[All, 1]]