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 mod_pow(a: i64, e:i64) ->i64{
    let mut ret = 1;
    if e ==0 {
        return 1;
    }
    ret = mod_pow(a * a % MOD, e / 2);
    if e % 2 == 1 {
        ret = ret * a % MOD;
    }
    ret
}

fn main() {
    let mut line = String::new();
    std::io::stdin().read_line(&mut line);
    let mut it = line.split_whitespace();
    let mut n:i64 = it.next().unwrap().parse().unwrap();
    let mut k:i64 = it.next().unwrap().parse().unwrap();
    let mut add = vec![0i64; k as usize + 1];
    let mut ans = 0;
    for i in 1..=k {
        let count = k / i;
        let sum = add[i as usize];
        ans = (ans + mod_pow(count, n) * (i - sum)) % MOD;
        add[i as usize] = i - sum;
        let mut j = i;
        while j <= k {
            add[j as usize] += i-sum;
            j += i;
        }
    }
    println!("{}", ans);
}