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.Collections.Generic;
using System.Linq;
using System.IO;
using SB = System.Text.StringBuilder;
//using System.Threading.Tasks;
//using System.Text.RegularExpressions;
//using System.Globalization;
//using System.Diagnostics;
using static System.Console;
using System.Numerics;
using static System.Math;
using pair = Pair<int, int>;

class Program
{
    static void Main()
    {
        //SetOut(new StreamWriter(OpenStandardOutput()) { AutoFlush = false });
        new Program().solve();
        Out.Flush();
    }
    readonly Scanner cin = new Scanner();
    readonly int[] dd = { 0, 1, 0, -1, 0 }; //→↓←↑
    readonly int mod = 1000000007;
    readonly int dom = 998244353;
    bool chmax<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) < 0) { a = b; return true; } return false; }
    bool chmin<T>(ref T a, T b) where T : IComparable<T> { if (b.CompareTo(a) < 0) { a = b; return true; } return false; }

    void solve()
    {
        int N = cin.nextint;
        int K = cin.nextint;

        var L = new List<int>();
        for (int i = 1; i * i <= N; i++)
        {
            int j = N / i;
            L.Add(i);
            if (i != j) L.Add(j);
        }
        //WriteLine(L.Count);
        L.Sort();

        var dp = new RangeAddFenwickTree(L.Count);
        //var before = new RangeAddFenwickTree(L.Count);
        dp.Add(0, 0, 1);

        for (int i = 0; i < K; i++)
        {
            //swap(ref dp, ref before);
            //dp.Clear();
            var next = new RangeAddFenwickTree(L.Count);

            int LAST = 0;
            for (int j = 0; j < L.Count; j++)
            {
                var value = dp[j, j] % mod;
                //int Y = N / L[j];
                //int id = L.lower_bound(Y);
                int id = L.Count - 1 - j;
                next.Add(0, id, value * (L[j] - LAST) % mod);
                LAST = L[j];
            }
            //int LAST = 0;
            //for (int j = 0; j < L.Count; j++)
            //{
            //    var P = dp[j, j] % mod;
            //    dp.Add(j, j, (L[j] - LAST - 1) * P % mod);
            //    LAST = L[j];
            //}
            //dp.Items.join();
            dp = next;
        }

        //WriteLine(dp[0, L.Count - 1] % mod);
        int last = 0;
        long ans = 0;
        for (int j = 0; j < L.Count; j++)
        {
            ans += dp[j, j] % mod * (L[j] - last) % mod;
            last = L[j];
        }
        WriteLine(ans % mod);
    }

    void swap<T>(ref T a, ref T b)
    {
        var t = a;
        a = b;
        b = t;
    }

}

/// <summary>
/// [0,<see cref="Mod"/>) までの値を取るような数
/// </summary>
/// <OriginalAuthor>camypaper</OriginalAuthor>
struct ModInt
{
    /// <summary>
    /// 剰余を取る値．
    /// </summary>
    public const long Mod = (int)1e9 + 7;

    /// <summary>
    /// 実際の数値．
    /// </summary>
    public long num;
    /// <summary>
    /// 値が <paramref name="n"/> であるようなインスタンスを構築します．
    /// </summary>
    /// <param name="n">インスタンスが持つ値</param>
    /// <remarks>パフォーマンスの問題上，コンストラクタ内では剰余を取りません．そのため，<paramref name="n"/> ∈ [0,<see cref="Mod"/>) を満たすような <paramref name="n"/> を渡してください．このコンストラクタは O(1) で実行されます．</remarks>
    public ModInt(long n) { num = n; }
    /// <summary>
    /// このインスタンスの数値を文字列に変換します．
    /// </summary>
    /// <returns>[0,<see cref="Mod"/>) の範囲内の整数を 10 進表記したもの．</returns>
    public override string ToString() { return num.ToString(); }
    public static ModInt operator +(ModInt l, ModInt r) { l.num += r.num; if (l.num >= Mod) l.num -= Mod; return l; }
    public static ModInt operator -(ModInt l, ModInt r) { l.num -= r.num; if (l.num < 0) l.num += Mod; return l; }
    public static ModInt operator *(ModInt l, ModInt r) { return new ModInt(l.num * r.num % Mod); }
    public static implicit operator ModInt(long n) { n %= Mod; if (n < 0) n += Mod; return new ModInt(n); }

    /// <summary>
    /// 与えられた 2 つの数値からべき剰余を計算します．
    /// </summary>
    /// <param name="v">べき乗の底</param>
    /// <param name="k">べき指数</param>
    /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
    public static ModInt Pow(ModInt v, long k) { return Pow(v.num, k); }

    /// <summary>
    /// 与えられた 2 つの数値からべき剰余を計算します．
    /// </summary>
    /// <param name="v">べき乗の底</param>
    /// <param name="k">べき指数</param>
    /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
    public static ModInt Pow(long v, long k)
    {
        long ret = 1;
        for (k %= Mod - 1; k > 0; k >>= 1, v = v * v % Mod)
            if ((k & 1) == 1) ret = ret * v % Mod;
        return new ModInt(ret);
    }
    /// <summary>
    /// 与えられた数の逆元を計算します．
    /// </summary>
    /// <param name="v">逆元を取る対象となる数</param>
    /// <returns>逆元となるような値</returns>
    /// <remarks>法が素数であることを仮定して，フェルマーの小定理に従って逆元を O(log N) で計算します．</remarks>
    public static ModInt Inverse(ModInt v) { return Pow(v, Mod - 2); }
}

class BinomialCoefficient
{
    public ModInt[] fact, ifact;
    /// <summary>
    /// <paramref name="n"/>は <paramref name="Mod"/>未満でお願いします。
    /// </summary>
    /// <param name="n"></param>
    public BinomialCoefficient(ModInt _n)
    {
        int n = (int)_n.num;
        fact = new ModInt[n + 1];
        ifact = new ModInt[n + 1];
        fact[0] = 1;
        for (int i = 1; i <= n; i++)
            fact[i] = fact[i - 1] * i;
        ifact[n] = ModInt.Inverse(fact[n]);
        for (int i = n - 1; i >= 0; i--)
            ifact[i] = ifact[i + 1] * (i + 1);
        ifact[0] = ifact[1];
    }
    public ModInt this[int n, int r]
    {
        get
        {
            if (n < 0 || n >= fact.Length || r < 0 || r > n) return 0;
            return fact[n] * ifact[n - r] * ifact[r];
        }
    }
    public ModInt RepeatedCombination(int n, int k)
    {
        if (k == 0) return 1;
        return this[n + k - 1, k];
    }
}

/// <OriginalAuthor>camypaper</OriginalAuthor>
class FenwickTree
{
    readonly int n;
    readonly long[] bit;
    readonly int max = 1;
    public FenwickTree(int size)
    {
        n = size; bit = new long[n + 1];
        while ((max << 1) <= n) max <<= 1;
    }
    public void Clear()
    {
        for (int i = 0; i < bit.Length; i++)
        {
            bit[i] = 0;
        }
    }
    /// <summary>sum[a, b] 0-indexed</summary>
    public long this[int i, int j] { get { i++; j++; return i <= j ? this[j] - this[i - 1] : 0; } }
    /// <summary>sum[1, i] 1-indexed</summary>
    public long this[int i] { get { long s = 0; for (; i > 0; i -= i & -i) s += bit[i]; return s; } }
    public int LowerBound(long w)
    {
        if (w <= 0) return 0;
        int x = 0;
        for (int k = max; k > 0; k >>= 1)
            if (x + k <= n && bit[x + k] < w)
            {
                w -= bit[x + k];
                x += k;
            }
        return x + 1;
    }
    /// <summary>add v to bit[i] 0-indexed</summary>
    public void Add(int i, long v)
    {
        i++;
        //if (i == 0) System.Diagnostics.Debug.Fail("BIT is 1 indexed");
        for (; i <= n; i += i & -i) bit[i] += v;
    }
    public long[] Items
    {
        get
        {
            var ret = new long[n];
            for (int i = 0; i < ret.Length; i++)
                ret[i] = this[i, i];
            return ret;
        }
    }
}

class RangeAddFenwickTree
{
    readonly int n;
    FenwickTree a, b;
    public RangeAddFenwickTree(int n)
    {
        this.n = n;
        a = new FenwickTree(n);
        b = new FenwickTree(n);
    }
    public void Clear()
    {
        a.Clear();
        b.Clear();
    }
    /// <summary>Add V to[i, j] 0-indexed</summary>
    public void Add(int i, int j, long v)
    {
        //a.Add(i, -(i - 1) * v); a.Add(j + 1, j * v);
        //b.Add(i, v); b.Add(j + 1, -v);
        a.Add(i, -i * v); a.Add(j + 1, (j + 1) * v);
        b.Add(i, v); b.Add(j + 1, -v);
    }
    /// <summary>Sum [1, i] 1-indexed</summary>
    public long this[int i] => a[i] + b[i] * i;
    /// <summary>Sum [i, j] 0-indexed</summary>
    public long this[int i, int j] { get { i++; j++; return i <= j ? this[j] - this[i - 1] : 0; } }
    public long[] Items
    {
        get
        {
            var ret = new long[n];
            for (int i = 0; i < ret.Length; i++)
                ret[i] = this[i, i];
            return ret;
        }
    }
}

static class Ex
{
    public static void join<T>(this IEnumerable<T> values, string sep = " ") => WriteLine(string.Join(sep, values));
    public static string concat<T>(this IEnumerable<T> values) => string.Concat(values);
    public static string reverse(this string s) { var t = s.ToCharArray(); Array.Reverse(t); return t.concat(); }

    public static int lower_bound<T>(this IList<T> arr, T val) where T : IComparable<T>
    {
        int low = 0, high = arr.Count;
        int mid;
        while (low < high)
        {
            mid = ((high - low) >> 1) + low;
            if (arr[mid].CompareTo(val) < 0) low = mid + 1;
            else high = mid;
        }
        return low;
    }
    public static int upper_bound<T>(this IList<T> arr, T val) where T : IComparable<T>
    {
        int low = 0, high = arr.Count;
        int mid;
        while (low < high)
        {
            mid = ((high - low) >> 1) + low;
            if (arr[mid].CompareTo(val) <= 0) low = mid + 1;
            else high = mid;
        }
        return low;
    }
}

class Pair<T, U> : IComparable<Pair<T, U>> where T : IComparable<T> where U : IComparable<U>
{
    public T f; public U s;
    public Pair(T f, U s) { this.f = f; this.s = s; }
    public int CompareTo(Pair<T, U> a) => f.CompareTo(a.f) != 0 ? f.CompareTo(a.f) : s.CompareTo(a.s);
    public override string ToString() => $"{f} {s}";
}

class Scanner
{
    string[] s; int i;
    readonly char[] cs = new char[] { ' ' };
    public Scanner() { s = new string[0]; i = 0; }
    public string[] scan => ReadLine().Split();
    public int[] scanint => Array.ConvertAll(scan, int.Parse);
    public long[] scanlong => Array.ConvertAll(scan, long.Parse);
    public double[] scandouble => Array.ConvertAll(scan, double.Parse);
    public string next
    {
        get
        {
            if (i < s.Length) return s[i++];
            string st = ReadLine();
            while (st == "") st = ReadLine();
            s = st.Split(cs, StringSplitOptions.RemoveEmptyEntries);
            i = 0;
            return next;
        }
    }
    public int nextint => int.Parse(next);
    public long nextlong => long.Parse(next);
    public double nextdouble => double.Parse(next);
}