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;

typedef long long LL;

const int mod=1e9+7;

map<LL,LL>mp;

LL n,k;

LL qsm(LL a,LL b) 
{
	LL res=1;
	while(b) {
		if(b&1) res=res*a%mod;
		b>>=1;
		a=a*a%mod;
	}
	return res;
}

int main()
{
	
	cin>>n>>k;
	
	for(int i=k;i>=1;i--) {
		LL res=0;
		res=qsm(k/i,n);
		int bei=2;
		while(bei*i<=k) res-=mp[bei*i],bei++;
		mp[i]=res%mod;
	}
	//cout<<"----"<<mp[1]<<'\n';
	LL sum=0;
	for(LL i=1;i<=k;i++) {
		sum=(sum+(LL)i*mp[i]%mod)%mod;
	}
	cout<<sum%mod<<'\n';
	
	return 0;
}

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