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> 
#define ll long long
#define F first
#define S second
#define FF first.first
#define FS first.second
#define pb push_back
using namespace std;
const ll N=1000006, INF=1e10, MOD=1000000007;
     
ll n, k, q=1, a[N], b[4][N], c[N], f[N], m, o=3, h, w, mx, mn, ans[N], Ans;
string s, S;
vector <ll> v[N];
pair<ll, ll> p[N];

ll POW(ll x, ll y){
	
	ll X=x;x=1;
	while(y){
		if(y%2)x*=X,x%=MOD,y--;
		y/=2, X*=X, X%=MOD;
	}
	return x;
}

main(){ios_base::sync_with_stdio(false), cin.tie(0);
                    
    cin>>n>>k;
	
	for (int i=k; i>=1; i--){
		ans[i]=POW((k/i),n);
		for (int j=2*i; j<=k; j+=i){
			ans[i]-=ans[j];
			if(ans[i]<0)ans[i]+=MOD;
		}
		(Ans+=(ans[i]*i))%=MOD;
	}
	
	cout<<Ans;
}