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# (Mono-mcs 6.8.0.105) (judge verdict: Accepted)

using System;
using System.Collections.Generic;

class Program
{
    static long mod = 1000000007;

    static long pow(long x, long k, long mod) {
        k %= mod-1;
        long r = 1;
        while(k>0) {
            if((k&1)>0) r = (r*x)%mod;
            k>>=1;
            x = x*x%mod;
        }
        return r;
    }
    
    static void Main()
    {
        var str = Console.ReadLine().Split(new char[] {' '});
        long N = long.Parse(str[0]), K = long.Parse(str[1]);

        var D = new List<long>[K+1];
        for(long i=1; i<=K; ++i) D[i] = new List<long>();

        var X = new long[K+1];
        
        for(long d=1; d<=K; ++d) {
            for(long c=2; c*d<=K; ++c) D[c*d].Add(d);
            X[d] = pow(K/d, N, mod); //(K/d)**N % mod;
        }

        long ans = 0;
        for(long d=K; d>=1; --d) {
            foreach(var i in D[d]) {
                X[i] -= X[d];
                X[i] %= mod;
            }
            ans += X[d]*d%mod;
        }

        Console.WriteLine(ans%mod);

    }


}