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)

static MOD: usize = 1000000007;
fn ipow(a: usize, n: usize) -> usize {
    let mut r = 1;
    let mut b = a;
    let mut k = n;
    while k>0 {
        if (k&1)!=0 {
            r = r*b%MOD;
        }
        b = b*b%MOD;
        k >>= 1;
    }
    r
}
fn main() {
    proconio::input! {
        n: usize,
        k: usize,
    }
    let mut c = Vec::new(); c.resize(k+1, 0);
    for i in (1..k+1).rev() {
        let m = k/i;
        c[i]=ipow(m,n);
        for j in 2..=m {
            let v = c[i*j];
            if c[i]<v {
                c[i]+=MOD;
            }
            c[i]-=v;
        }
    }
    let mut t=0;
    for i in 1..k+1 {
        t = (t+c[i]*i)%MOD;       
    }
    println!("{}",t);
}