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.Text;
using System.Collections.Generic;
using static System.Math;
using System.Linq;
using static System.Array;

namespace ConsoleApp2
{
    class Program
    {
        static long N;
        static long K;
        static long[] ans;
        static long mod = 1000000007;
        static void Main(string[] args)
        {
            var NK = Console.ReadLine().Split(' ').Select(x => long.Parse(x)).ToArray();
            N = NK[0];
            K = NK[1];

            ans = new long[K + 1];
            long tmp = 0;
            for (int i = (int)K; i >= 1; i--)
            {
                ans[i] = Cul(i);
                tmp += (i * ans[i])%mod;
                tmp %= mod;
            }
            Console.WriteLine(tmp);
        }

        static long Cul(int num)
        {
            //var res  = (long)Pow((K/num),N);
            var res = Modular.Pow((K / num),(int)N);
            for (long i = 2*num; i <= K; i += num)
            {
                res -= ans[i];
            }
            return res.value;
        }


        class Modular
        {
            private const int M = 1000000007;
            public long value;
            public Modular(long value) { this.value = value; }
            public static implicit operator Modular(long a)
            {
                var m = a % M;
                return new Modular((m < 0) ? m + M : m);
            }
            public static Modular operator +(Modular a, Modular b)
            {
                return (a.value + b.value) % M;
            }
            public static Modular operator -(Modular a, Modular b)
            {
                return a.value - b.value;
            }
            public static Modular operator *(Modular a, Modular b)
            {
                return a.value * b.value;
            }
            public static Modular Pow(Modular a, int n)
            {
                switch (n)
                {
                    case 0:
                        return 1;
                    case 1:
                        return a;
                    default:
                        var p = Pow(a, n / 2);
                        return p * p * Pow(a, n % 2);
                }
            }
            public static Modular operator /(Modular a, Modular b)
            {
                return a * Pow(b, M - 2);
            }
            private static readonly List<int> facs = new List<int> { 1 };
            private static Modular Fac(int n)
            {
                for (int i = facs.Count; i <= n; ++i)
                {
                    facs.Add((int)(Math.BigMul(facs.Last(), i) % M));
                }
                return facs[n];
            }
            public static Modular Ncr(int n, int r)
            {
                return (n < r) ? 0
                     : (n == r) ? 1
                                : Fac(n) / (Fac(r) * Fac(n - r));
            }
            public static explicit operator int(Modular a)
            {
                return (int)a.value;
            }

            public static Modular Lcm(Modular a, Modular b)
            {
                var tmp = Gcdl(a.value, b.value);
                return ((a * b) / new Modular(tmp));
            }
            static Modular Gcd(Modular m, Modular n)
            {
                if (n.value > m.value) return Gcd(n, m);
                if (n.value == 0) return m;
                var r = new Modular(m.value % n.value);
                return Gcd(n, r);
            }

            static long Gcdl(long m, long n)
            {
                if (n > m) return Gcdl(n, m);
                if (n == 0) return m;
                return Gcdl(n, m % n);
            }
        }
    }    
}