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};

fn pow<T>(mut x: T, mut y: T, p: T) -> T
where
    T: num::PrimInt,
{
    let mut ret = num::one();
    while y > num::zero() {
        if y & num::one() == num::one() {
            ret = ret * x % p;
        }
        x = x * x % p;
        y = y >> 1;
    }
    ret
}

#[fastout]
fn main() {
    input! {
        n: u64, k: u64
    }
    const M: u64 = 1000000007;
    let mut v = vec![0; (k + 1) as usize];
    let mut ans = 0;
    for x in (1..=k).rev() {
        let mut s = pow(k / x, n, M);
        for y in 2..=(k / x) {
            s = (s + (M - v[(x * y) as usize])) % M;
        }
        v[x as usize] = s;
        ans += s * x;
        ans %= M;
    }
    println!("{}", ans);
}