What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$?
1. First, we need to calculate the sum of the cubes \(3^3 + 4^3 + 5^3\):
   \[
   3^3 = 27, \quad 4^3 = 64, \quad 5^3 = 125
   \]
   \[
   3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216
   \]

2. We need to find the smallest positive integer \(k\) such that \(k \cdot 216 = a^n\) for some positive integers \(a\) and \(n\) with \(n > 1\).

3. Notice that \(216\) can be expressed as a perfect power:
   \[
   216 = 6^3
   \]

4. Since \(216\) is already a perfect cube, we can write:
   \[
   k \cdot 216 = a^n
   \]
   If we choose \(k = 1\), then:
   \[
   1 \cdot 216 = 216 = 6^3
   \]
   Here, \(a = 6\) and \(n = 3\), which satisfies the condition \(n > 1\).

5. Therefore, the smallest positive integer \(k\) that satisfies the given condition is \(k = 1\).

The final answer is \(\boxed{1}\)