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)

const MOD: i64 = 1_000_000_007;

fn main() {
    let (n, k): (usize, usize) = {
        let mut buf = String::new();
        std::io::stdin().read_line(&mut buf).unwrap();
        let mut iter = buf.split_whitespace();
        (
            iter.next().unwrap().parse().unwrap(),
            iter.next().unwrap().parse().unwrap(),
        )
    };

    let pow = |mut a: i64, mut n: usize| -> i64 {
        let mut res = 1;
        while n > 0 {
            if n & 1 > 0 {
                res = res * a % MOD;
            }
            a = a * a % MOD;
            n >>= 1;
        }
        res
    };

    let mut count: Vec<i64> = vec![0; k + 1];
    for i in (1..=k).rev() {
        count[i] = pow((k / i) as i64, n);
        let mut j = i + i;
        while j <= k {
            count[i] = (count[i] + MOD - count[j]) % MOD;
            j += i;
        }
    }

    let ans = (1..=k).fold(0, |acc, i| (acc + i as i64 * count[i]) % MOD);
    println!("{}", ans);
}