Well, I have the following code:

Clear["Global`*"];
a =;
b =;
\[Alpha] =;
\[Beta] =;
k =;
ParallelTable[
  If[TrueQ[Length[
      Select[PowersRepresentations[n, a, b], 
       DuplicateFreeQ[#] && ! MemberQ[#, 0] &]] >= k], n, 
   Nothing], {n, \[Alpha], \[Beta]}] //. {} -> Nothing

But for a given $a,b, k,\alpha$ and $\beta$ this calculation is really slow. Is there a way to speed things up a bit?

I am a student in elementary number theory and looking for power representations of certain numbers. For relatively small sets, it is convenient to use this code but for larger sets, it becomes way too slow.

Thanks for any advice and help.

Example runnable code:

Clear["Global`*"];
a = 3;
b = 5;
\[Alpha] = 500000;
\[Beta] = 10000000;
k = 6;
ParallelTable[
  If[TrueQ[Length[
      Select[PowersRepresentations[n, a, b], 
       DuplicateFreeQ[#] && ! MemberQ[#, 0] &]] >= k], n, 
   Nothing], {n, \[Alpha], \[Beta]}] //. {} -> Nothing