Solving a system of two equations of cube integers using ParallelTable

Question

Well, I have the following problem: I need to solve the following system of equations:

$$ \begin{cases} n=a^3+b^3+c^3\\ \\ n=d^3+k^3+f^3 \end{cases}\tag1 $$

Where:

• $n\ne a\ne b\ne c\ne d\ne k\ne f$;
• $1\le a<n,1\le b<n,1\le c<n,1\le d<n,1\le k<n,1\le f<n$;
• $n\in\mathbb{N}^+,a\in\mathbb{N}^+,b\in\mathbb{N}^+,c\in\mathbb{N}^+,d\in\mathbb{N}^+,k\in\mathbb{N}^+,f\in\mathbb{N}^+$

And I want to search for solutions using the following range: $2\le n\le1000^3$.

I thought of the following code:

ParallelTable[{n,Solve[{n==a^3+b^3+c^3,n==d^3+k^3+f^3,
1<=a<n&&1<=b<n&&1<=c<n&&1<=d<n&&1<=k<n&&1<=f<n&&a!=b!=c!=d!=k!=f},
{a,b,c,d,k,f},Integers]},{n,2,1000^3}]

Question: is there a faster way to let Mathematica evaluate this?

---

EDIT:

After seeing the answer that was posted by @kglr, I wondered can I use that technique to solve the same problem by using more variables:

$$ \begin{cases} n=a^3+b^3+c^3\\ \\ n=d^3+k^3+f^3\\ \\ n=g^3+h^3+m^3\\ \\ n=a^3+k^3+m^3\\ \\ n=g^3+k^3+c^3\\ \\ n=a^3+d^3+g^3\\ \\ n=b^3+k^3+h^3\\ \\ n=c^3+f^3+m^3 \end{cases}\tag2 $$

All the conditions are the same, so they all can not be equal to each other and needs to be an integer bigger than zero.

Answer 1

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