cp's OEIS Frontend

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.

%I A385565 #15 Jul 18 2025 14:46:46
%S A385565 11,21,64,144,330,846,342,252,1331,1008,720,1890,3780,729,4200,2016,
%T A385565 1000,216,6300,8352,10800,12312,8568,19440,8280,9576,21204
%N 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.
%C A385565 a(13) and a(15) not found up to k = 3300, a(14) = 729, a(16) = 2016, a(17) = 1000, a(18) = 216.
%e A385565 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.
%t A385565 s = Table[{k, Length@Select[PowersRepresentations[k^4, 3, 3],
%t A385565       0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 400}];
%t A385565 a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]
%Y A385565 Cf. A024975, A025419, A377444, A385354, A385566.
%K A385565 nonn,more
%O A385565 1,1
%A A385565 _Zhining Yang_, Jul 03 2025
%E A385565 a(13), a(15), a(19)-a(21) from _Chai Wah Wu_, Jul 08 2025
%E A385565 a(22)-a(27) from _Chai Wah Wu_, Jul 18 2025