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)

fn read_stdin() -> String {
    use std::io::Read;
    let mut input = String::new();
    std::io::stdin().read_to_string(&mut input).unwrap();
    input
}

fn scan<'a, I: Iterator<Item = &'a str>, T: std::str::FromStr>(tokens: &mut I) -> T {
    tokens.next().unwrap().parse().ok().unwrap()
}

const MODULO: usize = 1_000_000_007;

fn main() {
    let input = read_stdin();
    let tokens = &mut input.split_whitespace();
    let n: usize = scan(tokens);
    let k: usize = scan(tokens);
    let mut ans: usize = 0;
    for mut x in 1..=k {
        let c = power(k / x, n);
        let mut a = 1;
        let mut d = 2;
        while d * d <= x {
            if x % d == 0 {
                a = a * (d - 1) % MODULO;
                x /= d;
                while x % d == 0 {
                    a = a * d % MODULO;
                    x /= d;
                }
            }
            d += 1;
        }
        if x != 1 {
            a = a * (x - 1) % MODULO;
        }
        ans = (ans + a * c) % MODULO;
    }
    println!("{}", ans);
}

fn power(mut b: usize, mut e: usize) -> usize {
    let mut a = 1;
    while e != 0 {
        if e & 1 != 0 {
            a = a * b % MODULO;
        }
        b = b * b % MODULO;
        e >>= 1;
    }
    a
}