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 MOD: usize = 1_000_000_007;

fn pow(mut x: usize, mut e: usize) -> usize {
    let mut res = 1;
    while e > 0 {
        if e & 1 == 1 {
            res = res * x % MOD;
        }
        x = x * x % MOD;
        e >>= 1;
    }
    res
}

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

    let mut memo: Vec<usize> = vec![0; k + 1];
    for x in (1..=k).rev() {
        let mut c = pow(k / x, n);

        for i in 2..=k / x {
            if let Some(&v) = memo.get(i * x) {
                c = (c + MOD - v) % MOD;
            }
        }

        memo[x] = c;
    }

    let mut ans = 0;
    for x in 1..=k {
        ans = (ans + memo[x] * x % MOD) % MOD;
    }

    println!("{}", ans);
}