Problem:
Score : 900 points

Problem Statement

For a positive integer n, let us define f(n) as the number of digits in base 10.

You are given an integer S.
Count the number of the pairs of positive integers (l, r) (l \leq r) such that f(l) + f(l + 1) + ... + f(r) = S, and find the count modulo 10^9 + 7.

Constraints

1 \leq S \leq 10^8

Input

Input is given from Standard Input in the following format:

S

Output

Print the answer.

Sample Input 1

1

Sample Output 1

9

There are nine pairs (l, r) that satisfies the condition: (1, 1), (2, 2), ..., (9, 9).

Sample Input 2

2

Sample Output 2

98

There are 98 pairs (l, r) that satisfies the condition, such as (1, 2) and (33, 33).

Sample Input 3

123

Sample Output 3

460191684

Sample Input 4

36018

Sample Output 4

966522825

Sample Input 5

1000

Sample Output 5

184984484

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

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



class Myon
{
    public Myon() { }
    public static int Main()
    {
        new Myon().calc();
        return 0;
    }


    Scanner cin;

    long mod = 1000000007;

    void calc()
    {
        cin = new Scanner();
        int S = cin.nextInt();
        int sum = 0;
        long ans = 0;
        int start = 1;
        int add = 0;
        int target = 1;
        int dec = 1;
        int targetd = 10;
        for (int end = 1; start < 1e8; end++)
        {
            if(end == target)
            {
                add++;
                target *= 10;
            }
            sum += add;
            while(sum > S)
            {
                sum -= dec;
                start++;
                if (start == targetd)
                {
                    dec++;
                    targetd *= 10;
                }
            }
            if (sum == S && start < 1e8) ans++;
        }
        ans %= mod;

        for (int i = 1; i * i <= S; i++)
        {
            int k = i;
            int kk = S / i;
            if (k * kk != S) continue;
            if (k >= 9) {
                ans += (9 * powmod(10, k - 1)) - (S / k - 1);
                ans %= mod;
            }
            if (k == kk) continue;
            k = kk;

            if (k >= 9)
            {
                ans += (9 * powmod(10, k - 1)) - (S / k - 1);
                ans %= mod;
            }
        }

        /*
        for (int k = 9; k <= S; k++)
        {
            if(S % k == 0)
            {
                ans += (nowNum - (S / k - 1));
                ans %= mod;
            }
            nowNum *= 10;
            nowNum %= mod;
        }
        */

        for (int k = 1; k <= S; k++)
        {
            int div = S / k;
            if (div < 9) break;
            if (div * k == S) continue;
            ans++;
        }


        ans += mod;
        ans %= mod;
        Console.WriteLine(ans);
    }

    
    long powmod(long a, long p)
    {
        long ans = 1;
        long mul = a;
        for (; p > 0; p >>= 1, mul = (mul * mul) % mod)
        {
            if ((p & 1) == 1) ans = (ans * mul) % mod;
        }
        return ans;
    }

}


class Scanner
{
    string[] s;
    int i;

    char[] cs = 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(cs, StringSplitOptions.RemoveEmptyEntries);
        if (s.Length == 0) return next();
        i = 0;
        return s[i++];
    }

    public int nextInt()
    {
        return int.Parse(next());
    }
    public int[] ArrayInt(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 nextLong()
    {
        return long.Parse(next());
    }

    public long[] ArrayLong(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 nextDouble()
    {
        return double.Parse(next());
    }


    public double[] ArrayDouble(int N, double add = 0)
    {
        double[] Array = new double[N];
        for (int i = 0; i < N; i++)
        {
            Array[i] = nextDouble() + add;
        }
        return Array;
    }
}