Steps to One

2000ms

262144K

Description:

Vivek initially has an empty array $$$a$$$ and some integer constant $$$m$$$.

He performs the following algorithm:

Select a random integer $$$x$$$ uniformly in range from $$$1$$$ to $$$m$$$ and append it to the end of $$$a$$$.
Compute the greatest common divisor of integers in $$$a$$$.
In case it equals to $$$1$$$, break
Otherwise, return to step $$$1$$$.

Find the expected length of $$$a$$$. It can be shown that it can be represented as $$$\frac{P}{Q}$$$ where $$$P$$$ and $$$Q$$$ are coprime integers and $$$Q\neq 0 \pmod{10^9+7}$$$. Print the value of $$$P \cdot Q^{-1} \pmod{10^9+7}$$$.

Input:

The first and only line contains a single integer $$$m$$$ ($$$1 \leq m \leq 100000$$$).

Output:

Print a single integer — the expected length of the array $$$a$$$ written as $$$P \cdot Q^{-1} \pmod{10^9+7}$$$.

Sample Input:

Sample Output:

Sample Input:

Sample Output:

Sample Input:

Sample Output:

333333338

Note:

In the first example, since Vivek can choose only integers from $$$1$$$ to $$$1$$$, he will have $$$a=[1]$$$ after the first append operation, and after that quit the algorithm. Hence the length of $$$a$$$ is always $$$1$$$, so its expected value is $$$1$$$ as well.

In the second example, Vivek each time will append either $$$1$$$ or $$$2$$$, so after finishing the algorithm he will end up having some number of $$$2$$$'s (possibly zero), and a single $$$1$$$ in the end. The expected length of the list is $$$1\cdot \frac{1}{2} + 2\cdot \frac{1}{2^2} + 3\cdot \frac{1}{2^3} + \ldots = 2$$$.

Informação

Provedor Codeforces

Código CF1139D