SeedRandom[1] 
    While[Length[pr] < 2, 
      pr = Select[FreeQ @ 0] @ PowersRepresentations[RandomInteger[{2, 10^6}], 3, 3]]; //
         AbsoluteTiming // First

>     5.8*10^-6


     {{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}