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::{fastout, input};

const M: i128 = 1e+9 as i128 + 7;

#[fastout]
fn main() {
    input! {in_n: usize, in_k: usize};
    let mut cnt = vec![0i128; in_k + 1];

    let mut n = in_n;
    let mut mul = Vec::new();

    while n != 0 {
        if n % 2 == 1 {
            mul.push(true);
            n -= 1;
        } else {
            mul.push(false);
        }

        n /= 2;
    }

    for k in 1..=in_k {
        let mut p = (in_k / k) as i128;
        let mut v = 1i128;

        for m in &mul {
            if *m {
                v = (v * p) % M;
            }
            p = (p * p) % M;
        }

        cnt[k] = v;
    }

    for k in (1..=in_k).rev() {
        for j in ((k * 2)..=in_k).step_by(k) {
            cnt[k] = (cnt[k] - cnt[j] + M) % M;
        }
    }

    let mut count = 0i128;

    for k in 1..=in_k {
        count = (count + cnt[k] * k as i128) % M;
    }

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