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;

const int mod = 1e9 + 7;

long long power(long long p, int e){
  long long ret = 1;
  while(e){
    if(e & 1){
      ret *= p;
      ret %= mod;
      e--;
    }
    p *= p;
    p %= mod;
    e >>= 1;
  }
  
  return ret;
}

int main(){
  int N, K;
  cin >> N >> K;
  
  long long c[K + 1];
  for(int i = K; i > 0; i--){
    c[i] = power(K / i, N);
    
    for(int j = (i << 1); j <= K; j += i){
      c[i] -= c[j];
      c[i] %= mod;
    }
  }
  
  long long sum = 0;
  for(int i = 1; i <= K; i++){
    sum += c[i] * i % mod;
    sum %= mod;
  }
  
  if(sum < 0) sum += mod;
  
  cout << sum << endl;
  
  return 0;
}