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

fn main() {
    input! {
        n: usize,
        k: usize,
    }
    let mut result = 0;
    let mut dp = vec![0; k + 1];
    for i in (1..=k).rev() {
        dp[i] = mod_pow(k / i, n, MODULO);
        for j in 2.. {
            if j * i > k {
                break;
            }
            dp[i] = (MODULO + dp[i] - dp[j * i]) % MODULO;
        }
        result = (result + dp[i] * i) % MODULO;
    }

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