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)

#region いつもの
using System;
using System.Collections.Generic;
using System.Linq;
using IEnumerable = System.Collections.IEnumerable;
using IEnumerator = System.Collections.IEnumerator;
using Unsafe = System.Runtime.CompilerServices.Unsafe;
using BigInteger = System.Numerics.BigInteger;
using TextReader = System.IO.TextReader;
using StringBuilder = System.Text.StringBuilder;
using static Global;
using static NumGlobal;

static class Global
{
    public static T[] NewArray<T>(int len0, T value) => new T[len0].Fill(value);
    public static T[] NewArray<T>(int len0, Func<T> factory)
    {
        var arr = new T[len0];
        for (int i = 0; i < arr.Length; i++) arr[i] = factory();
        return arr;
    }
    public static T[][] NewArray<T>(int len0, int len1, T value) where T : struct
    {
        var arr = new T[len0][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, value);
        return arr;
    }
    public static T[][] NewArray<T>(int len0, int len1, Func<T> factory)
    {
        var arr = new T[len0][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, factory);
        return arr;
    }
    public static T[][][] NewArray<T>(int len0, int len1, int len2, T value) where T : struct
    {
        var arr = new T[len0][][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, value);
        return arr;
    }
    public static T[][][] NewArray<T>(int len0, int len1, int len2, Func<T> factory)
    {
        var arr = new T[len0][][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, factory);
        return arr;
    }
    public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, T value) where T : struct
    {
        var arr = new T[len0][][][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, value);
        return arr;
    }
    public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, Func<T> factory)
    {
        var arr = new T[len0][][][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, factory);
        return arr;
    }
    private class ComparerReverseImpl<T> : Comparer<T> where T : IComparable<T> { public override int Compare(T y, T x) => x.CompareTo(y); public override bool Equals(object obj) => obj != null && GetType() == obj.GetType(); public override int GetHashCode() => GetType().GetHashCode(); }
    public static IComparer<T> ComparerReverse<T>() where T : IComparable<T> => new ComparerReverseImpl<T>();
    public static string AllLines<T>(IEnumerable<T> source) => string.Join("\n", source);
    public static string AllJoin<T>(IEnumerable<T> source) => string.Join(" ", source);
}
static class NumGlobal
{
    public static int Pow(int x, int y)
    {
        int res = 1;
        for (; y > 0; y >>= 1)
        {
            if ((y & 1) == 1) res *= x;
            x *= x;
        }
        return res;
    }
    public static BigInteger ParseBigInteger(ReadOnlySpan<char> s)
    {
        // 自前実装の方が速い
        BigInteger res;
        if (s.Length % 9 == 0)
            res = 0;
        else
        {
            res = new BigInteger(int.Parse(s.Slice(0, s.Length % 9)));
            s = s.Slice(s.Length % 9);
        }

        while (s.Length > 0)
        {
            var sp = s.Slice(0, 9);
            res *= 1000_000_000;
            res += int.Parse(sp);
            s = s.Slice(9);
        }
        return res;
    }
    public static int BitCount(int x) { x -= (x >> 1) & 0x55555555; x = (x & 0x33333333) + ((x >> 2) & 0x33333333); x = (x + (x >> 4)) & 0x0f0f0f0f; x += x >> 8; x += x >> 16; return x & 0x3f; }
    public static int BitCount(long x) { x -= (x >> 1) & 0x5555555555555555; x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333); x = (x + (x >> 4)) & 0x0f0f0f0f0f0f0f0f; x += x >> 8; x += x >> 16; x += x >> 32; return (int)(x & 0x0000007f); }
    public static int MSB(int x) { x |= x >> 1; x |= x >> 2; x |= x >> 4; x |= x >> 8; x |= x >> 16; return BitCount(x) - 1; }
    public static int MSB(long x) { x |= x >> 1; x |= x >> 2; x |= x >> 4; x |= x >> 8; x |= x >> 16; x |= x >> 32; return BitCount(x) - 1; }
    public static int LSB(int x) { x |= x << 1; x |= x << 2; x |= x << 4; x |= x << 8; x |= x << 16; return 32 - BitCount(x); }
    public static int LSB(long x) { x |= x << 1; x |= x << 2; x |= x << 4; x |= x << 8; x |= x << 16; x |= x << 32; return 64 - BitCount(x); }
}
static class Ext
{
    public static T[] Fill<T>(this T[] arr, T value)
    {
        Array.Fill(arr, value);
        return arr;
    }
    public static T[] Sort<T>(this T[] arr) { Array.Sort(arr); return arr; }
    public static string[] Sort(this string[] arr) => Sort(arr, StringComparer.OrdinalIgnoreCase);
    public static T[] Sort<T, U>(this T[] arr, Func<T, U> selector) where U : IComparable<U> => Sort(arr, (a, b) => selector(a).CompareTo(selector(b)));
    public static T[] Sort<T>(this T[] arr, Comparison<T> comparison) { Array.Sort(arr, comparison); return arr; }
    public static T[] Sort<T>(this T[] arr, IComparer<T> comparer) { Array.Sort(arr, comparer); return arr; }
    public static T[] Reverse<T>(this T[] arr) { Array.Reverse(arr); return arr; }
    public static (int index, T max) MaxBy<T>(this T[] arr) where T : IComparable<T>
    {
        T max = arr[0];
        int maxIndex = 0;
        for (int i = 0; i < arr.Length; i++)
        {
            if (max.CompareTo(arr[i]) < 0)
            {
                max = arr[i];
                maxIndex = i;
            }
        }
        return (maxIndex, max);
    }
    public static (TSource item, TMax max) MaxBy<TSource, TMax>
        (this IEnumerable<TSource> source, Func<TSource, TMax> maxBySelector)
        where TMax : IComparable<TMax>
    {
        TMax max;
        TSource maxByItem;
        var e = source.GetEnumerator();
        e.MoveNext();
        maxByItem = e.Current;
        max = maxBySelector(maxByItem);
        while (e.MoveNext())
        {
            var item = e.Current;
            var next = maxBySelector(item);
            if (max.CompareTo(next) < 0)
            {
                max = next;
                maxByItem = item;
            }
        }
        return (maxByItem, max);
    }
    public static (int index, T min) MinBy<T>(this T[] arr) where T : IComparable<T>
    {
        T min = arr[0];
        int minIndex = 0;
        for (int i = 0; i < arr.Length; i++)
        {
            if (min.CompareTo(arr[i]) > 0)
            {
                min = arr[i];
                minIndex = i;
            }
        }
        return (minIndex, min);
    }
    public static (TSource item, TMin min) MinBy<TSource, TMin>
        (this IEnumerable<TSource> source, Func<TSource, TMin> minBySelector)
        where TMin : IComparable<TMin>
    {
        TMin min;
        TSource minByItem;

        var e = source.GetEnumerator();
        e.MoveNext();
        minByItem = e.Current;
        min = minBySelector(minByItem);
        while (e.MoveNext())
        {
            var item = e.Current;
            var next = minBySelector(item);
            if (min.CompareTo(next) > 0)
            {
                min = next;
                minByItem = item;
            }
        }
        return (minByItem, min);
    }
    public static IComparer<T> Reverse<T>(this IComparer<T> comparer) => Comparer<T>.Create((x, y) => comparer.Compare(y, x));
    public static Dictionary<TKey, int> GroupCount<TSource, TKey>(this IEnumerable<TSource> source, Func<TSource, TKey> keySelector) => source.GroupBy(keySelector).ToDictionary(g => g.Key, g => g.Count());
    public static Dictionary<TKey, int> GroupCount<TKey>(this IEnumerable<TKey> source) => source.GroupCount(i => i);
    public static TValue Get<TKey, TValue>(this Dictionary<TKey, TValue> dic, TKey key)
    {
        dic.TryGetValue(key, out var v);
        return v;
    }
    public static TValue GetOrInit<TKey, TValue>(this Dictionary<TKey, TValue> dic, TKey key, TValue value)
    {
        if (dic.TryGetValue(key, out var v))
            return v;
        return dic[key] = value;
    }
}
class ΔDebugView<T> { private IEnumerable<T> collection; public ΔDebugView(IEnumerable<T> collection) { this.collection = collection ?? throw new ArgumentNullException(nameof(collection)); }[System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.RootHidden)] public T[] Items => collection.ToArray(); }
#pragma warning disable CA1819
[System.Diagnostics.DebuggerNonUserCode]
public class ConsoleReader { private string[] ReadLineSplit() => textReader.ReadLine().Split(Array.Empty<char>(), StringSplitOptions.RemoveEmptyEntries); private string[] line = Array.Empty<string>(); private int linePosition; private TextReader textReader; public ConsoleReader(TextReader tr) { textReader = tr; } public int Int => int.Parse(String); public int Int0 => Int - 1; public long Long => long.Parse(String); public double Double => double.Parse(String); public string String { get { if (linePosition >= line.Length) { linePosition = 0; line = ReadLineSplit(); } return line[linePosition++]; } } public class SplitLine { public SplitLine(ConsoleReader cr) { String = cr.ReadLineSplit(); cr.line = Array.Empty<string>(); } public int[] Int => String.Select(x => int.Parse(x)).ToArray(); public int[] Int0 => String.Select(x => int.Parse(x) - 1).ToArray(); public long[] Long => String.Select(x => long.Parse(x)).ToArray(); public double[] Double => String.Select(x => double.Parse(x)).ToArray(); public string[] String { get; } } public SplitLine Split => new SplitLine(this); public class RepeatReader { ConsoleReader cr; int count; public RepeatReader(ConsoleReader cr, int count) { this.cr = cr; this.count = count; } public T[] Select<T>(Func<ConsoleReader, T> factory) { var arr = new T[count]; for (int i = 0; i < count; i++) arr[i] = factory(cr); return arr; } public string[] String => this.Select(cr => cr.String); public int[] Int => this.Select(cr => cr.Int); public int[] Int0 => this.Select(cr => cr.Int - 1); public long[] Long => this.Select(cr => cr.Long); public double[] Double => this.Select(cr => cr.Double); } public RepeatReader Repeat(int count) => new RepeatReader(this, count); }
public class Program
{
    [System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.Never)] private ConsoleReader cr;

    public Program(ConsoleReader consoleReader) { this.cr = consoleReader; }
    static void Main(string[] args) => Console.WriteLine(new Program(new ConsoleReader(args.Length > 0 ? new System.IO.StreamReader(args[0]) : Console.In)).Calc());

    #endregion
    public object Calc()
    {
        var N = cr.Int;
        var K = cr.Int;

        var dp = new Mod[K + 1];
        for (int i = K; i > 0; i--)
        {
            var baisu = K / i;
            dp[i] = Mod.Pow(baisu, N);

            for (int j = i * 2; j < dp.Length; j += i)
                dp[i] -= dp[j];
        }

        Mod res = 0;
        for (int i = 1; i < dp.Length; i++)
        {
            res += dp[i] * i;
        }
        return res;
    }
    int Gcd(params int[] nums)
    {
        var gcd = nums[0];
        for (var i = 1; i < nums.Length; i++)
            gcd = Gcd(nums[i], gcd);
        return gcd;
    }
    int Gcd(int a, int b) => b > a ? Gcd(b, a) : (b == 0 ? a : Gcd(b, a % b));
}
readonly struct Mod : IEquatable<Mod>
{
    public const long mod = 1000000007;
    public static readonly Mod invalid = new Mod(-1, false);
    public readonly long val;
    private Mod(long val, bool isValid) : this(val) { if (!isValid) this.val = val; }
    public Mod(long val) { this.val = val % mod; if (this.val < 0) this.val += mod; }
    public override bool Equals(object obj) => (obj is Mod) ? this == ((Mod)obj) : false;
    public bool Equals(Mod obj) => this == obj;
    public override int GetHashCode() => val.GetHashCode();
    public override string ToString() => val.ToString();
    public static implicit operator Mod(long x) => new Mod(x);
    public static Mod operator +(Mod x, Mod y) => (x.val + y.val) % mod;
    public static Mod operator -(Mod x, Mod y) => x.val >= y.val ? (x.val - y.val) % mod : (x.val - y.val) % mod + mod;
    public static Mod operator *(Mod x, Mod y) => (x.val * y.val) % mod;
    public static Mod operator /(Mod x, Mod y) => x * y.Inverse();
    public static bool operator ==(Mod x, Mod y) => x.val == y.val;
    public static bool operator !=(Mod x, Mod y) => x.val != y.val;
    public Mod Inverse()
    {
        long a = val, b = mod, u = 1, v = 0;
        while (b > 0)
        {
            long t = a / b;
            var b2 = a - t * b;
            a = b; b = b2;
            var v2 = u - t * v;
            u = v; v = v2;
        }
        u %= mod;
        if (u < 0) u += mod;
        return u;
    }
    public static Mod Pow(Mod x, int y)
    {
        Mod res = 1;
        for (; y > 0; y >>= 1)
        {
            if ((y & 1) == 1) res *= x;
            x *= x;
        }
        return res;
    }

    public static Factor CreateFactor(int max) => new Factor(max);
    public class Factor
    {
        private readonly Mod[] fac, finv;
        public Factor(int max)
        {
            ++max;
            var inv = new Mod[max];
            fac = new Mod[max]; finv = new Mod[max];
            fac[0] = fac[1] = 1;
            finv[0] = finv[1] = 1;
            inv[1] = 1;
            for (var i = 2; i < max; i++)
            {
                fac[i] = fac[i - 1] * i;
                inv[i] = mod - inv[mod % i].val * (mod / i) % mod;
                finv[i] = finv[i - 1] * inv[i];
            }
        }

        // 二項係数計算
        public Mod Combine(int n, int k)
        {
            if (n < k) return 0;
            if (n < 0 || k < 0) return 0;
            return fac[n] * finv[k] * finv[n - k];
        }

        public Mod Factorial(int n) => fac[n];
        public Mod FactorialInvers(int n) => finv[n];
    }
}