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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Numerics;

namespace AtCoder.Problems
{
    public class ProblemE
    {
        public static void Main(string[] args)
        {
            var s = new SolveE();
            s.Solve();
        }

        public class Scanner
        {
            private string[] _s;
            private int _i;
            private readonly char[] _separators = new char[] {' '};

            public Scanner()
            {
                _s = new string[0];
                _i = 0;
            }

            public string Next()
            {
                if (_i < _s.Length) return _s[_i++];
                string st = Console.ReadLine();
                while (st == "") st = Console.ReadLine();
                _s = st.Split(_separators, StringSplitOptions.RemoveEmptyEntries);
                if (_s.Length == 0) return Next();
                _i = 0;
                return _s[_i++];
            }

            public int NextInt() => int.Parse(Next());
            public long NextLong() => long.Parse(Next());
            public ulong NextULong() => ulong.Parse(Next());
            public double NextDouble() => double.Parse(Next());

            public int[] GetIntArray(int n, int add = 0)
            {
                int[] array = new int[n];
                for (int i = 0; i < n; i++)
                {
                    array[i] = NextInt() + add;
                }
                return array;
            }

            public long[] GetLongArray(int n, long add = 0)
            {
                long[] array = new long[n];
                for (int i = 0; i < n; i++)
                {
                    array[i] = NextLong() + add;
                }
                return array;
            }

            public double[] GetDoubleArray(int n, double add = 0)
            {
                double[] array = new double[n];
                for (int i = 0; i < n; i++)
                {
                    array[i] = NextDouble() + add;
                }
                return array;
            }
        }
    }

    public class SolveE
    {
        public void Solve()
        {
            var cin = new ProblemE.Scanner();
            var n = cin.NextInt();
            var k = cin.NextInt();

            var result = GetAnswer(n, k);
            Console.WriteLine(result);
        }

        public long GetAnswer(int n, int k)
        {
            const int mod = 1000000007;
            var d = new long[k + 1];
            for (int i = 1; i < d.Length; i++)
            {
                d[i] = (long)BigInteger.ModPow(k / i, n, mod);
            }

            for (int i = k; i > 0; i--)
            {
                for (int j = 2 * i; j <= k; j += i)
                {
                    d[i] -= d[j];
                    d[i] = (d[i] + mod) % mod;
                }
            }
            var ans = 0L;
            for (int i = 1; i < d.Length; i++)
            {
                ans += d[i] * i;
                ans %= mod;
            }
            return ans;
        }
    }
}