Find a function $f(n)$ on the positive integers with positive integer values such that $f( f(n) ) = 1993 n^{1945}$ for all $n$.
1. **Define the function and the problem constraints:**
   We need to find a function \( f(n) \) on the positive integers with positive integer values such that \( f(f(n)) = 1993 n^{1945} \) for all \( n \).

2. **Construct the chains:**
   We start with the positive integers \( 1, 2, \ldots \) and sieve through the numbers by taking the smallest number \( n \) still in the list. We remove \( n, f^2(n), f^4(n), \ldots \) from the list. This list of numbers is called a "chain".

3. **Prove that any two chains are disjoint:**
   Suppose two chains shared a value. Applying \( f^{-1} \circ f^{-1} \) to that value will output the value of the previous number in each chain. Repeating this process until we get to one of the chains' beginnings shows that that chain's beginning was already used in a previous chain via the sieve, which is a contradiction to the construction. Therefore, any two chains are disjoint.

4. **Pair up the chains arbitrarily:**
   Let \( a_1, a_2, \ldots \) be one chain and \( b_1, b_2, \ldots \) be the chain it is paired with. We define \( f(a_i) = b_i \) and \( f(b_i) = a_{i+1} \) for all \( i \geq 1 \).

5. **Verify the function definition:**
   - For \( f(a_i) = b_i \):
     \[
     f(f(a_i)) = f(b_i) = a_{i+1}
     \]
   - For \( f(b_i) = a_{i+1} \):
     \[
     f(f(b_i)) = f(a_{i+1}) = b_{i+1}
     \]

6. **Check the consistency with the problem statement:**
   We need \( f(f(n)) = 1993 n^{1945} \). By the construction, we have:
   \[
   f(f(a_i)) = a_{i+1} \quad \text{and} \quad f(f(b_i)) = b_{i+1}
   \]
   This implies that the function \( f \) is consistent with the problem statement, as every number shows up in some chain, and all positive integers are successfully mapped.

7. **Conclusion:**
   The function \( f \) defined by the pairing of chains satisfies the given condition \( f(f(n)) = 1993 n^{1945} \).

The final answer is \( \boxed{ f(n) } \) as defined by the pairing of chains.