SeedRandom[1]
While[Length[pr = Select[FreeQ @ 0] @
PowersRepresentations[RandomInteger[{2, 10^6}], 3, 3]] < 2]; //
AbsoluteTiming // First
0.0009008
{{a, b, c}, {d, k, f}, n} = Join[pr[[;;2]], {Total[pr[[1]]^3]}]
{{2, 3, 16}, {3, 9, 15}, 4131}
Total[pr^3, {2}]
{4131, 4131}