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: Wrong Answer)

#include <bits/stdc++.h>
#define rep(i,a,b) for(int i=int(a);i<=int(b);++i)
using namespace std;
template<class A>void pr(A a){cout << (a) << endl;}
template<class A,class B>void pr(A a,B b){cout << a << " "  ;pr(b);}
template<class A,class B,class C>void pr(A a,B b,C c){cout << a << " " ;pr(b,c);}
template<class A,class B,class C,class D>void pr(A a,B b,C c,D d){cout << a << " " ;pr(b,c,d);}
typedef long long int ll;
typedef pair<ll, ll> l_l;
typedef pair<int, int> i_i;
int INF=numeric_limits<int>::max();
// for(int j=0;j<N;++j) {

ll modpow(ll a, ll n, ll MOD) {
  ll res = 1;
  while(n>0){
    if(n&1) res = res * a % MOD;
    a = a * a % MOD; // a を 2乗したものに置き換える。（2進数だと1桁上がる）
    n >>= 1; // 2進数で1桁下げる
  }
  return res;
}

int main(void) {
  int n,k;cin>>n>>k;
  const int MOD=pow(10,9)+7;
  ll res[k+20];
  ll sum=0;
  for(int i=k;i>0;--i) {
    int t = k/i;
    res[i-1]=modpow(t,n,MOD);
    rep(j,2,t) {
      res[i-1]-=res[j*i-1];
    }
    sum+=i*res[i-1];
    // pr(res[i-1],i,t);
  }
  pr(sum);
  return 0;
}