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;

const int N=1e5+20,MOD=1e9+7;

int n,k,dp[N],ans;

int mul(int a,int b) {return (1ll*a*b)%MOD;}

int power(int a,int b)
{
	int num=a,ans=1;
	while(b)
	{
		if(b&1) ans=mul(ans,num);
		b>>=1;
		num=mul(num,num);
	}
	return ans;
}

int main()
{
	cin>>n>>k;

	for(int i=k;i>=1;i--)
	{
		int choices=1;
		for(int j=2*i;j<=k;j+=i,choices++)
		{
			dp[i]=(dp[i]-dp[j])%MOD;
			if(dp[i]<0) dp[i]+=MOD;
		}
		dp[i]=(dp[i]+power(choices,n))%MOD;
		ans=(ans+mul(i,dp[i]))%MOD;
	}

	printf("%d",ans);
}