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: u64 = 1_000_000_000 + 7;

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

fn main() {
    input! {
        n: u64,
        k: usize,
    }

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

    for i in (1..=k).rev() {
        v[i] = pow((k / i) as u64, n);
        let mut j = i * 2;
        while j <= k {
            v[i] = (v[i] + MOD - v[j]) % MOD;
            j += i;
        }
        res += v[i] * i as u64 % MOD;
        res %= MOD;
    }

    println!("{}", res)
}