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::input;


fn pow(a: u64, b: u64) -> u64 {
  let m = 1_000_000_007u64;
  if b == 0  { return 1; }
  if b % 2 == 1 {
    return a * pow(a, b-1) % m;
  } else {
    let r = pow(a, b/2);
    return r * r % m;
  }
}

fn main() {
  input! {
    n: u64, 
    k: u64,
  }
  let m = 1_000_000_007u64;
  let mut v: Vec<u64> = Vec::<u64>::with_capacity((k+1) as usize);
  for i in 0..k+1 { v.push(0); }
  let mut res: u64 = 0;

  for i in (1..k+1).rev(){
    v[i as usize] = pow(k / i, n);
    let mut j: usize = i as usize*2 as usize;
    while j <= k as usize {
      v[i as usize] = (v[i as usize] + m - v[j]) % m;
      j += i as usize;
    }
    res += v[i as usize] * i as u64 % m;
    res %= m;
  }
  println!("{}", res);
}