Problem:
Score : 500 points

Problem Statement

Consider sequences \{A_1,...,A_N\} of length N consisting of integers between 1 and K (inclusive).

There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.

Since this sum can be enormous, print the value modulo (10^9+7).

Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.

Constraints

2 \leq N \leq 10^5

1 \leq K \leq 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K

Output

Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).

Sample Input 1

3 2

Sample Output 1

9

\gcd(1,1,1)+\gcd(1,1,2)+\gcd(1,2,1)+\gcd(1,2,2)
+\gcd(2,1,1)+\gcd(2,1,2)+\gcd(2,2,1)+\gcd(2,2,2)
=1+1+1+1+1+1+1+2=9

Thus, the answer is 9.

Sample Input 2

3 200

Sample Output 2

10813692

Sample Input 3

100000 100000

Sample Output 3

742202979

Be sure to print the sum modulo (10^9+7).

Solution in Rust (1.42.0) (judge verdict: Accepted)

use proconio::input;

fn mod_pow(a: u64, p: u64, m: u64) -> u64 {
    if p == 0 {
        1
    } else {
        let half = mod_pow(a, p/2, m);
        let res = half * half % m;
        if p % 2 == 0 { res } else  { res * a % m }
    }
}

fn main() {
    input! {
        n : u64,
        k : usize
    };
    let mut a = [0; 100001];
    let m : u64 = 1000000007;
    let mut res : u64 = 0;
    for i in (1..=k).rev() {
        a[i] = mod_pow((k/i) as u64, n, m);
        let mut j = 2 * i;
        while j <= k {
            a[i] = (a[i] + m - a[j]) % m;
            j += i;
        }
        res = (res + a[i] * i as u64) % m;
    }
    println!("{}", res);
}