MathNet 2011 Question 15

Given integers $n \ge 3$ and $k \ge 1$, find all positive integers $a_1 \lt a_2 \lt \dots \lt a_n$ such that $\frac{1}{a_1^k} + \frac{1}{a_2^k} + \dots + \frac{1}{a_n^k} = 1$.