CF653G Move by Prime

Pussycat Sonya has an array consisting of $ n $ positive integers. There are $ 2^{n} $ possible subsequences of the array. For each subsequence she counts the minimum number of operations to make all its elements equal. Each operation must be one of two: - Choose some element of the subsequence and multiply it by some prime number. - Choose some element of the subsequence and divide it by some prime number. The chosen element must be divisible by the chosen prime number. What is the sum of minimum number of operations for all $ 2^{n} $ possible subsequences? Find and print this sum modulo $ 10^{9}+7 $ .

The first line of the input contains a single integer $ n $ ( $ 1

Print the sum of minimum number of operation for all possible subsequences of the given array modulo $ 10^{9}+7 $ .

In the first sample, there are $ 8 $ possible subsequences: $ (60,60,40) $ , $ (60,60) $ , $ (60,40) $ , $ (60,40) $ , $ (60) $ , $ (60) $ , $ (40) $ and $ () $ (empty subsequence). For a subsequence $ (60,60,40) $ we can make all elements equal by two operations — divide $ 40 $ by $ 2 $ to get $ 20 $ , and then multiply $ 20 $ by $ 3 $ to get $ 60 $ . It's impossible to achieve the goal using less operations and thus we add $ 2 $ to the answer. There are two subsequences equal to $ (60,40) $ and for each of them the also need to make at least $ 2 $ operations. In each of other subsequences all numbers are already equal, so we need $ 0 $ operations for each of them. The sum is equal to $ 2+2+2=6 $ .