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 std::io::{Read, stdin};

use proconio::input;

fn main() {
    const modulo: u64 = 1000000007;
    input! {
        n: u64,
        k: usize
    }

    let mut gcd_count = vec![0; k + 1];
    let mut res = 0u64;

    for gcd in (1..=k).rev() {
        gcd_count[gcd] = pow((k / gcd) as u64, n);
        for i in 2..=k / gcd {
            gcd_count[gcd] = (gcd_count[gcd] + modulo - gcd_count[gcd * i]) % modulo;
        }
        res += gcd_count[gcd] * gcd as u64 % modulo;
        res %= modulo;
    }

    println!("{:?}", res);
}

fn pow(a: u64, b: u64) -> u64 {
    const modulo: u64 = 1000000007;
    if b == 0 {
        1
    } else if b % 2 == 1 {
        a * pow(a, b - 1) % modulo
    } else {
        let r = pow(a, b / 2);
        r * r % modulo
    }
}