CF1436F Sum Over Subsets

You are given a multiset $ S $ . Over all pairs of subsets $ A $ and $ B $ , such that: - $ B \subset A $ ; - $ |B| = |A| - 1 $ ; - greatest common divisor of all elements in $ A $ is equal to one; find the sum of $ \sum_{x \in A}{x} \cdot \sum_{x \in B}{x} $ , modulo $ 998\,244\,353 $ .

The first line contains one integer $ m $ ( $ 1 \le m \le 10^5 $ ): the number of different values in the multiset $ S $ . Each of the next $ m $ lines contains two integers $ a_i $ , $ freq_i $ ( $ 1 \le a_i \le 10^5, 1 \le freq_i \le 10^9 $ ). Element $ a_i $ appears in the multiset $ S $ $ freq_i $ times. All $ a_i $ are different.

Print the required sum, modulo $ 998\,244\,353 $ .

A multiset is a set where elements are allowed to coincide. $ |X| $ is the cardinality of a set $ X $ , the number of elements in it. $ A \subset B $ : Set $ A $ is a subset of a set $ B $ . In the first example $ B=\{1\}, A=\{1,2\} $ and $ B=\{2\}, A=\{1, 2\} $ have a product equal to $ 1\cdot3 + 2\cdot3=9 $ . Other pairs of $ A $ and $ B $ don't satisfy the given constraints.