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 int64_t MOD=1e9+7;
 
int64_t modpow(int64_t a,int64_t n){
  int64_t res=1;
  if(n==0) return res;
  while(n>0){
    if(n&1) res=(res*a)%MOD;
    a=(a*a)%MOD;
    n >>= 1;
  }
  return res;
}
 
int main(){
  int64_t N,K;
  cin>>N>>K;
  int64_t ans=0;
 
  
  //i|gcdなるような総数mod[i]
  vector<int64_t> mod(K+1,0);
  for(int i=1;i<=K;++i){
    mod[i]=modpow(K/i,N);
  }
  
  //i==gcdなるような総数dp[i]
  vector<int64_t> dp(K+1,0);
  for(int64_t i=K;i>=1;--i){
    dp[i]=mod[i];
    for(int64_t j=2;i*j<=K;++j){
      dp[i] += MOD-dp[i*j];
    }
    dp[i] %= MOD;
    ans += i*dp[i];
    ans %= MOD;
  }
 
  cout<<ans<<endl;
}