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;

const MODULO: usize = 1_000_000_007;

fn pow(a: usize, n: usize, p: usize) -> usize {
    if n == 0 {
        1
    } else if n % 2 == 0 {
        let x = pow(a, n/2, p);
        x * x % p
    } else {
        let x = pow(a, n/2, p);
        a * ((x * x) % p) % p
    }
}

fn main() {
    input! {
        n: usize,
        k: usize,
    }

    let mut count = vec![0; k];
    let mut ans = 0;

    for x in (1..=k).rev() {
        let c = pow(k/x, n, MODULO);
        count[x-1] = c;
        for i in 2.. {
            if i * x  > k {
                break;
            }
            count[x-1] = (count[x-1] + MODULO - count[i*x-1]) % MODULO;
        }
        ans += x * count[x-1] % MODULO;
        ans %= MODULO;
    }
    
    println!("{}", ans);
}