Generate n-ary numbers
Jelly, 12 bytes

Æf*³&lt;‘Ạ
1Ç#Ṫ


Takes n and k (one-indexed) as command-line arguments.

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How it works

1Ç#Ṫ     Main link. Left argument: n. Right argument: k

1        Set the return value to 1.
 Ç#      Execute the helper link above for r = 1, 2, 3, ... until k of them return
         a truthy value. Yield the list of all k matches.
   Ṫ     Tail; extract the last match.


Æf*³&lt;‘Ạ  Helper link. Argument: r

Æf       Compute all prime factors of r.
  *³     Elevate them to the n-th power.
    &lt;‘   Compare all powers with r + 1.
      Ạ  All; return 1 if all comparisons were true, 0 if one or more were not.