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: Runtime Error)

#include<stdio.h>
#include<stdlib.h>
#include<math.h>
typedef long long ll;

ll gcd(ll a,ll b)
{
  if(b==0){
    return a;
  }else{
    return gcd(b,a%b);
  }
}

void main(void)
{
  ll N,K;
  ll p=1000000007;
  scanf("%lld %lld",&N,&K);
  ll g[K+1][K+1];
  for(ll i=1;i<=K;i++){
    for(ll j=1;j<=K;j++){
      g[i][j]=0;
    }
  }
  for(ll i=1;i<=K;i++){
    for(ll j=1;j<=K;j++){
      if(i>=j){
        g[i][gcd(i,j)]++;
      }else{
        g[i][gcd(j,i)]++;
      }
    }
  }
  ll T[N+1][K+1];
  for(ll i=1;i<=K;i++){
    T[1][i]=1;
  }
  ll sum=0;
  for(ll i=2;i<=N;i++){
    for(ll j=1;j<=K;j++){
      for(ll m=j;m<=K;m++){
        sum+=T[i-1][m]*g[m][j];
      }
      T[i][j]=sum;
      sum=0;
    }
  }
  ll ans=0;
  for(ll i=i;i<=K;i++){
    ans+=(i*T[N][i])%p;
  }
  
  printf("%lld",ans);

}