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;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;

internal partial class Solver {
    public void Run() {
        var n = ni();
        var k = ni();

        var mod = ModInt.SetModValue(1000000007);
        ModInt ans = 0;
        var dp = new ModInt[k + 1];
        for (int g = k; g >= 1; g--) {
            long cand = (k - g) / g + 1;
            ModInt pattern = ModInt.Pow(cand, n);
            for (int x = g + g; x <= k; x += g) {
                pattern -= dp[x];
            }
            dp[g] = pattern;
            ans += pattern * g;
        }

        cout.WriteLine(ans);
    }
}


public struct ModInt : IEquatable<ModInt> {
    private static int mod = 0;
    private long _value;

    public ModInt(long x) {
        _value = x;
        Normalize();
    }

    private static long RegularMod(long x, int mod) {
        if (x >= mod) {
            if (x < 2 * mod) {
                return x - mod;
            }

            return x % mod;
        }
        if (x >= 0) {
            return x;
        }

        x = mod - RegularMod(-x, mod);
        if (x == mod) {
            return 0;
        }

        return x;
    }

    public static int SetModValue(int m) {
        return mod = m;
    }

    private void Normalize() {
        _value = RegularMod(_value, mod);
    }

    public override string ToString() {
        return _value.ToString(CultureInfo.InvariantCulture);
    }

    public int ToInt() {
        return (int)_value;
    }

    public static bool operator ==(ModInt c1, ModInt c2) {
        return c1._value == c2._value;
    }

    public static bool operator !=(ModInt c1, ModInt c2) {
        return !(c1 == c2);
    }

    public static ModInt operator +(ModInt x, ModInt y) {
        return new ModInt(x._value + y._value);
    }
    public static ModInt operator -(ModInt x, ModInt y) {
        return new ModInt(x._value - y._value);
    }
    public static ModInt operator -(ModInt x) {
        return new ModInt(-x._value);
    }
    public static ModInt operator *(ModInt x, ModInt y) {
        return new ModInt(x._value * y._value);
    }
    public static ModInt operator /(ModInt x, ModInt y) {
        return new ModInt(x._value * Inverse(y._value, mod));
    }
    public static ModInt operator ++(ModInt x) {
        return x + 1;
    }
    public static ModInt operator --(ModInt x) {
        return x - 1;
    }
    public static ModInt Pow(ModInt x, long n) {
        ModInt r = 1;
        while (n > 0) {
            if ((n & 1) != 0) {
                r *= x;
            }

            x *= x;
            n >>= 1;
        }
        return r;
    }
    private static long ExtendedGcd(long a, long b, ref long x, ref long y) {
        if (b == 0) {
            x = 1; y = 0;
            return a;
        } else {
            var d = ExtendedGcd(b, a % b, ref y, ref x);
            y -= a / b * x;
            return d;
        }
    }

    private static long Inverse(long a, long mod) {
        long x = 0, y = 0;
        if (ExtendedGcd(a, mod, ref x, ref y) == 1) {
            return (x + mod) % mod;
        } else {
            throw new Exception("Invalid inverse " + a + " " + mod);
        }
    }

    public static implicit operator ModInt(long x) {
        return new ModInt(x);
    }

    public override bool Equals(object obj) {
        if (obj == null) {
            return false;
        }
        return _value.Equals(((ModInt)obj)._value);
    }

    public override int GetHashCode() {
        return _value.GetHashCode();
    }

    public bool Equals(ModInt other) {
        return _value.Equals(other._value);
    }
}

// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main() {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

internal static class LinqPadExtension {
    [Conditional("DEBUG")]
    public static void Dump<T>(this T obj) {
#if DEBUG
        LINQPad.Extensions.Dump(obj);
#endif
    }
}

public class Scanner {
    private readonly TextReader Reader;
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    private readonly char[] buffer = new char[2 * 1024];
    private int cursor = 0, length = 0;
    private string Token;
    private readonly StringBuilder sb = new StringBuilder(1024);

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return checked((int)NextLong()); }
    public long NextLong() {
        var s = Next();
        long r = 0;
        int i = 0;
        bool negative = false;
        if (s[i] == '-') {
            negative = true;
            i++;
        }
        for (; i < s.Length; i++) {
            r = r * 10 + (s[i] - '0');
#if DEBUG
            if (!char.IsDigit(s[i])) throw new FormatException();
#endif
        }
        return negative ? -r : r;
    }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (Token == null) {
            if (!StockToken()) {
                throw new Exception();
            }
        }
        var token = Token;
        Token = null;
        return token;
    }

    public bool HasNext() {
        if (Token != null) {
            return true;
        }

        return StockToken();
    }

    private bool StockToken() {
        while (true) {
            sb.Clear();
            while (true) {
                if (cursor >= length) {
                    cursor = 0;
                    if ((length = Reader.Read(buffer, 0, buffer.Length)) <= 0) {
                        break;
                    }
                }
                var c = buffer[cursor++];
                if (33 <= c && c <= 126) {
                    sb.Append(c);
                } else {
                    if (sb.Length > 0) break;
                }
            }

            if (sb.Length > 0) {
                Token = sb.ToString();
                return true;
            }

            return false;
        }
    }
}