2021-22 Sum of fractions

Determine all rational numbers that can be written as
\[
\frac{1}{n_1} + \frac{1}{n_1 n_2} + \frac{1}{n_1 n_2 n_3} + \dots + \frac{1}{n_1 n_2 n_3 \dots n_k} ,
\]
where \( n_1, n_2, n_3 \dots, n_k \) are positive integers greater than \(1\).

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