CF439E Devu and Birthday Celebration

Today is Devu's birthday. For celebrating the occasion, he bought $ n $ sweets from the nearby market. He has invited his $ f $ friends. He would like to distribute the sweets among them. As he is a nice guy and the occasion is great, he doesn't want any friend to be sad, so he would ensure to give at least one sweet to each friend. He wants to celebrate it in a unique style, so he would like to ensure following condition for the distribution of sweets. Assume that he has distributed $ n $ sweets to his friends such that $ i^{th} $ friend is given $ a_{i} $ sweets. He wants to make sure that there should not be any positive integer $ x>1 $ , which divides every $ a_{i} $ . Please find the number of ways he can distribute sweets to his friends in the required way. Note that the order of distribution is important, for example \[1, 2\] and \[2, 1\] are distinct distributions. As the answer could be very large, output answer modulo $ 1000000007 $ $ (10^{9}+7) $ . To make the problem more interesting, you are given $ q $ queries. Each query contains an $ n $ , $ f $ pair. For each query please output the required number of ways modulo $ 1000000007 $ $ (10^{9}+7) $ .

N/A

N/A

For first query: $ n=6,f=2 $ . Possible partitions are \[1, 5\] and \[5, 1\]. For second query: $ n=7,f=2 $ . Possible partitions are \[1, 6\] and \[2, 5\] and \[3, 4\] and \[4, 3\] and \[5, 3\] and \[6, 1\]. So in total there are 6 possible ways of partitioning.