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)

#![allow(unused_imports)]
use proconio::*;
use num::Integer;

const MAX_K: usize = 100005;
const MOD: i64 = 1000000007;
fn mod_pow(mut a:i64, mut n:usize)->i64 {
    let mut x = 1;
    while n > 0 {
        if n&1 == 1 {
            x *= a;
            x %= MOD;
        }
        a *= a;
        a %= MOD;
        n >>= 1;
    }
    x
}

fn main() {
   input! {
       n: usize,
       k: i64
   }

   let mut dp = [0i64; MAX_K];
   let mut ans: i64 = 0;
   for i in (1..=k).rev() {
       dp[i as usize] = mod_pow(k/i, n);
       for j in 2..=(k/i) as usize {
           dp[i as usize] -= dp[(i as usize)*j];
           if dp[i as usize] < 0 {
               dp[i as usize] += MOD;
           }
       }
       ans += i*dp[i as usize];
       ans %= MOD;
   }
   println!("{}", ans);
}