cp's OEIS Frontend

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.

%I A383877 #38 Jun 14 2025 17:58:48
%S A383877 14,13,55,26,52,63,70,66,56,104,102,143,161,91,117,112,78,236,180,217,
%T A383877 198,192,140,292,216,259,156,196,344,168,210,264,325,252,406,360,380,
%U A383877 402,315,338,234,308,351,182,396,408,399,432,441,312,474,636,513,273,336,476,618,666
%N 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.
%C A383877 The largest term for k<=10000 is a(3569)=9828.
%C A383877 Conjecture: a(n) != 0 for all n.
%H A383877 Zhining Yang, <a href="/A383877/b383877.txt">Table of n, a(n) for n = 1..1280</a>
%e A383877 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.
%t A383877 s=Table[{k,Length@Select[PowersRepresentations[k^3,4,3],0<#[[1]]<#[[2]]<#[[3]]<#[[4]]&]},{k,100}];
%t A383877 a=Table[SelectFirst[s,#[[2]]==k&],{k,9}][[All,1]]
%Y A383877 Cf. A377444, A384439.
%K A383877 nonn
%O A383877 1,1
%A A383877 _Zhining Yang_, May 13 2025