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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}

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}

If you want the 6 numbers all distinct:

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

 0.0000275

pr

 {{5, 53, 88}, {12, 29, 93}, {15, 74, 75}}

Total[pr[[1]]^3]

830474

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}

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}

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

{{2, 3, 16}, {3, 9, 15}}

Total[pr^3, {2}]

{4131, 4131}

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}