x^3 + y^3 + z^3 = k^n恰好有n组解的高效算法

northwolves

A383689  a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^n, where 0 < x < y < z has exactly n integer solutions.

DATA
36, 188, 54, 144, 90, 63, 66

COMMENTS
a(9)=15.

EXAMPLE
a(5)=90, because 90^5 =95^3 + 321^3 + 1804^3 = 186^3 + 1272^3 + 1566^3=704^3 + 1230^3 + 1546^3 = 720^3 + 810^3 + 1710^3 = 745^3 + 1335^3 + 1460^3 and no integer less than 90 has 5 solutions.

刚发现我提供的a(7)=66是错误的，{{242,7260,17182},{2892,3480,17532},{3672,10284,16284},{4048,6743,17193},{4356,4356,17424},{4565,6028,17259},{8789,12595,14058}} ,存在x=y的情况，所以66^7只有6组解符合要求。已知a(7)>120。

northwolves

A383879
a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^n, where 0 < x < y < z < w has exactly n integer solutions.

DATA
100, 42, 55, 34, 74

EXAMPLE
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.

MATHEMATICA
f[n_]:=Do[v=Select[PowersRepresentations[k^n, 4, 3], 0<#[[1]]<#[[2]]<#[[3]]<#[[4]]&]; d={n, k, v, Length@v}; If[d[[-1]]==n, Return[d]], {k, 100}]; Do[Print[f[n]], {n, 4}]