Devu and Birthday Celebration

题意翻译

一共有$q$个询问，每次询问给出两个正整数数$n,f$，要求输出有多少个不同的长度为$f$的序列$a_i$满足 - $\sum_{i=1}^f a_i=n$ - $gcd(a_1,a_2,\cdots ,a_f)=1$ 答案对$10^9+7$取模 数据范围：$1\leq q\leq 10^5,1\leq f\leq n\leq 10^5$

题目描述

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) $ .

输入输出格式

输入格式

The first line contains an integer $ q $ representing the number of queries $ (1<=q<=10^{5}) $ . Each of the next $ q $ lines contains two space space-separated integers $ n $ , $ f $ $ (1<=f<=n<=10^{5}) $ .

输出格式

For each query, output a single integer in a line corresponding to the answer of each query.

输入输出样例

输入样例 #1

5
6 2
7 2
6 3
6 4
7 4

输出样例 #1

2
6
9
10
20

说明

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.