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 mod 1000000007
#define maxN 100005
using namespace std;

long long p(long long x,long long y)
{
    if(!y) return 1ll;
    long long t=p(x,y/2)%mod;
    if(y%2) return (t*t%mod)*x%mod;
    else    return (t*t%mod);
}
long long r[maxN],u[maxN];
int nu,n,k;

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

    long long ngt=1;
    for(int i=1;i<=k;++i)
    {
        if(k%i==0) u[++nu]=i;
    }
    for(int i=1;i<=n;++i) ngt=ngt*i%mod;



    for(int i=2;i<=nu;++i)
    {
        long long ans=1,sl=k/u[i];
        for(long long gt=n*(sl-1)+1;gt<=n*sl;++gt) ans=(ans*gt)%mod;
        r[sl]=(ans*p(ngt,mod-2))%mod;
    }


    long long N=p(k,n);
    long long sum=0;
    for(int i=2;i<=k;++i)
    {
        int sl=k/i;
        N=(N-r[sl]+mod)%mod;
        sum=(sum+i*r[sl])%mod;
    }
    cout<<(N+sum)%mod;
}