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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
using lint = long long int;
const lint MOD = 1e9+7;

lint modpow(lint m, lint n){
  if(n == 0) return 1;
  else if(n % 2 == 0) return modpow(m * m % MOD, n/2);
  else{
    return modpow(m, n - 1) * m % MOD;
  }
}

signed main(){
	lint N, K; cin >> N >> K;
	lint ans = 0;
	vector<lint> data(K + 1, 0);
	for(lint i = K; i > 0; i--){
		lint num = modpow(K/i, N);
		for(lint j = 2 * i; j <= K; j += i){
			num = num - data[j] + MOD;
			num %= MOD;
		}
		data[i] = num;
		ans += num * i;
		ans %= MOD;
	}
	cout << ans << endl;
}