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;

const MOD: i64 = 1000000007;

fn mpow(a: i64, b: i64) -> i64 {
  let tmp = if b % 2 == 1 { a } else { 1 };
  match b {
    0 => 1,
    _ => tmp * mpow(a * a % MOD, b / 2) % MOD
  }
}

fn main() {
  input! {
    n: i64,
    k: usize
  }
  let mut dp = vec![0; k + 1];
  
  for i in (1..=k).rev() {
    dp[i] = mpow((k / i) as i64, n);
    for j in (i*2..=k).step_by(i) {
      dp[i] = (dp[i] + MOD - dp[j]) % MOD;
    }
  }
  
  println!("{}", (1..=k).map(|t| dp[t] * t as i64 % MOD).fold(0, |v, t| (v + t) % MOD));
}