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A386988 a(n) is the smallest integer w such that the equation x^2 + y^4 + z^6 = w^8 where GCD(x,y,z)=1 has exactly n positive integer solutions.

%I A386988 #10 Aug 12 2025 17:32:46
%S A386988 25,9,17,53,3
%N A386988 a(n) is the smallest integer w such that the equation x^2 + y^4 + z^6 = w^8 where GCD(x,y,z)=1 has exactly n positive integer solutions.
%C A386988 a(6)>1024.
%H A386988 Fang, <a href="https://www.zhihu.com/question/1921951729479586497">How to find non-trivial positive integer solutions to the Diophantine equation a^2 + b^4 + c^6 = d^8, where gcd(a,b,c) = 1?</a>.
%e A386988 a(3) = 17 because 17^8 = 36840^2 + 273^4 + 20^6 = 82367^2 + 24^4 + 24^6 = 48^2 + 287^4  + 24^6 and for no integer smaller than 17 we have 3 solutions.
%t A386988 f[w_]:=(v={};c=0;w8=w^8;
%t A386988 Do[yy=w8-z^6;Do[xx=yy-y^4;x=Sqrt@xx;
%t A386988 If[IntegerQ@x,If[GCD[x,y,z]==1,AppendTo[v,{x,y,z}];c++]],{y,Floor[yy^(1/4)]}],{z,Floor[w8^(1/6)]}];{c,w,v});
%t A386988 s=Table[{},5];
%t A386988 For[k=1,k<=60,k++,r=f[k][[1]];If[s[[r]]=={},s[[r]]=f[k];Print[s[[r]]]]]
%Y A386988 Cf. A386373.
%K A386988 nonn,more
%O A386988 1,1
%A A386988 _Zhining Yang_, Aug 12 2025