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;
static const int64_t mod=1000000007;
int64_t POW(int64_t x,int64_t y){
if(y==0)
  return 1;
  else if(y%2==0)
    return (POW(x,y/2)*POW(x,y/2))%mod;
  else
    return ((((x%mod)*POW(x,y/2))%mod)*POW(x,y/2))%mod;
}
int main(){
int64_t N,K;
  cin>>N>>K;
  vector<int64_t>ans(K+1);
  for(int i=K;1<=i;i--){
  int64_t a=POW(K/i,N);
    for(int j=2;j<=K/i;j++){
    a=(a-ans.at(i*j)+mod)%mod;
    }ans.at(i)=a;
  
  }int64_t A=0;
  for(int i=1;i<=K;i++){
  A=(A+ans.at(i)*i)%mod;
  }cout<<A<<endl;
   return 0;
}