Let $ b_i$, $i = 1,2,\ldots,n$, be any distinct positive integers, and $ P = \operatorname{rad}\left (\operatorname{lcm}\limits_{S \in \{1,2,\ldots,n\}}\left [\sum_{i\in S}b_i\right ]\right )$. Take $ a_i = Pb_i$.

Simpler, let $ b_i$, $i = 1,2,\ldots,n$, be any distinct positive integers, and $ P = \prod_{S \in \{1,2,\ldots,n\}}\sum_{i\in S}b_i$. Take $ a_i = Pb_i$.