Let $a_1, a_2, \ldots, a_{18}$ be a list of prime numbers such that $\frac{1}{64} \times a_1 \times a_2 \times \cdots \times a_{18}$ is one million. Determine the sum of all positive integers $n$ such that $$\sum_{i=1}^{18} \frac{1}{\log_{a_i} n}$$ is a positive integer.

Let's think step by step and output the final answer within \boxed{}.
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