Problem:
Score : 600 points

Problem Statement

Find the number of sequences of length K consisting of positive integers such that the product of any two adjacent elements is at most N, modulo 10^9+7.

Constraints

1\leq N\leq 10^9

1 2\leq K\leq 100 (fixed at 21:33 JST)

N and K are integers.

Input

Input is given from Standard Input in the following format:

N K

Output

Print the number of sequences, modulo 10^9+7.

Sample Input 1

3 2

Sample Output 1

5

(1,1), (1,2), (1,3), (2,1), and (3,1) satisfy the condition.

Sample Input 2

10 3

Sample Output 2

147

Sample Input 3

314159265 35

Sample Output 3

457397712

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.IO;
using System.Collections.Generic;
using System.Globalization;
using System.Linq;
using System.Text;
using E = System.Linq.Enumerable;

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

        var candidate = new List<int>();

        for (int i = 1; i * i <= n; i++) {
            candidate.Add(i);
            candidate.Add(i + 1);
            candidate.Add(n / i + 1);
            candidate.Add(n / i);
        }
        candidate = candidate.Distinct().ToList();
        candidate.Sort();

        var nextState = candidate
            .Select(c => LowerBound(candidate, n / c))
            .ToArray();

        const int mod = 1000000000 + 7;
        // dp[i,j] = 最後が i の場合の長さ j の配列の場合の数
        // dp[i,j] = sum dp[k,j-1] where i*k <= n
        var dp = new int[candidate.Count];
        for (int i = 0; i < candidate.Count; i++) {
            if (candidate[i] <= n) dp[i] = 1;
        }
        var next = new int[candidate.Count];
        var sumDp = new int[candidate.Count];
        for (int i = 0; i < K - 1; i++) {
            for (int j = 0; j + 1 < candidate.Count; j++) {
                var val = 0L;
                if (j - 1 >= 0) val += sumDp[j - 1];
                var count = candidate[j + 1] - candidate[j];
                val += 1L * count * dp[j];
                sumDp[j] = (int)(val % mod);
            }

            for (int j = 0; j + 1 < candidate.Count; j++) {
                next[j] = sumDp[nextState[j]];
            }
            dp = next;
        }
        long ans = 0;
        for (int i = 0; i + 1 < candidate.Count; i++) {
            var count = candidate[i + 1] - candidate[i];
            ans = (ans + 1L * count * dp[i]) % mod;
        }
        ans %= mod;
        cout.WriteLine(ans);
    }
    public static int LowerBound<T>(List<T> A, T x) { return LowerBound(A, 0, A.Count, x); }
    public static int LowerBound<T>(List<T> A, T x, IComparer<T> comparer) { return LowerBound(A, 0, A.Count, x, comparer); }
    public static int LowerBound<T>(List<T> A, T x, Comparison<T> comp) { return LowerBound(A, 0, A.Count, x, Comparer<T>.Create(comp)); }
    public static int LowerBound<T>(List<T> A, int start, int end, T x) { return LowerBound(A, start, end, x, Comparer<T>.Default); }
    public static int LowerBound<T>(List<T> A, int start, int end, T x, Comparison<T> comp) { return LowerBound(A, start, end, x, Comparer<T>.Create(comp)); }
    public static int LowerBound<T>(List<T> A, int start, int end, T x, IComparer<T> comparer) {
        int lower = start, upper = end - 1;
        int result = end;
        while (lower <= upper) {
            int pos = (lower + upper) / 2;
            if (comparer.Compare(x, A[pos]) <= 0) {
                result = pos;
                upper = pos - 1;
            } else {
                lower = pos + 1;
            }
        }
        return result;
    }

}

// PREWRITEN CODE BEGINS FROM HERE
partial class Solver : Scanner {
    public static void Main(string[] args) {
#if LOCAL
        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
    }

    public void RunMulti() {
        while (HasNext()) Run();
    }

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

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        this.cin = reader;
        this.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
}

public static class LinqPadExtension {
    static public T Dump<T>(this T obj) {
#if LOCAL
        return LINQPad.Extensions.Dump(obj);
#else
        return obj;
#endif
    }
}
public class Scanner {
    private readonly TextReader Reader;
    private readonly Queue<string> TokenQueue = new Queue<string>();
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

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

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

    public int NextInt() { return int.Parse(Next(), ci); }
    public long NextLong() { return long.Parse(Next(), ci); }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        var array = new string[size];
        for (int i = 0; i < size; i++) array[i] = Next();
        return array;
    }
    public int[] NextIntArray(int size) {
        var array = new int[size];
        for (int i = 0; i < size; i++) array[i] = NextInt();
        return array;
    }

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

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

    public string Next() {
        if (TokenQueue.Count == 0) {
            if (!StockTokens()) throw new InvalidOperationException();
        }
        return TokenQueue.Dequeue();
    }

    public bool HasNext() {
        if (TokenQueue.Count > 0)
            return true;
        return StockTokens();
    }
    static readonly char[] _separator = new[] { ' ' };
    private bool StockTokens() {
        while (true) {
            var line = Reader.ReadLine();
            if (line == null) return false;
            var tokens = line.Split(_separator, StringSplitOptions.RemoveEmptyEntries);
            if (tokens.Length == 0) continue;
            foreach (var token in tokens)
                TokenQueue.Enqueue(token);
            return true;
        }
    }
}