OK. The first approach that came to my mind was inclusion-exclusion over common elements in the set: Add one for every pair of indices $$$(i, j)$$$, subtract one for every tuple $$$(\{x\}, i, j)$$$ with $$$x \in A_i \cap A_j$$$, add one for every tuple $$$(\{x, y\}, i, j)$$$ with $$$x \neq y$$$ and $$$\{x, y\} \subseteq A_i \cap A_j$$$, and so on. But it is clear that only the sets $$$S$$$ which are subsets of at least one $$$A_i$$$ can contribute to the sum, and there are at most $$$2^7 \cdot 10^5$$$ such sets, and the contribution of each to the sum can be maintained using a hash-map: Process the sets in the input one at a time, and for each subset $$$S \subseteq A_i$$$, add or subtract as appropriate the number of input sets $$$A_j$$$ already encountered with $$$S \subseteq A_j$$$.

However, with sets of common prime factors of numbers less than $$$10^6$$$, there is a natural and obvious perfect hash function: the product, and so an array can be used instead.