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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.

%I A385409 #11 Jul 08 2025 18:37:22
%S A385409 10,42,39,153,126,276,273,312,315,476,588,336,546,777,1053,756,1216,
%T A385409 1386,1560,1134,1323,1488,1365,1368,1344,1596,2366,2496,2988,1680,
%U A385409 2548,1736,2184,3003,3720,2520,3185,3552,2268,3564,4095,3213,4578,4392,5208,4004,4599,5733
%N 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.
%C A385409 Conjecture: a(n) exists for all n.
%H A385409 Zhining Yang, <a href="/A385409/b385409.txt">Table of n, a(n) for n = 1..1341</a>
%e A385409 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.
%t A385409 s = Table[{k, Length@Select[PowersRepresentations[k^2, 4, 3],
%t A385409       0 < #[[1]] < #[[2]] < #[[3]] < #[[4]] &]}, {k, 500}];
%t A385409 a = Table[SelectFirst[s, #[[2]] == k &], {k, 10}][[All, 1]]
%Y A385409 Cf. A024975, A025419, A377444, A384430, A385354.
%K A385409 nonn
%O A385409 1,1
%A A385409 _Zhining Yang_, Jun 27 2025