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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)


public class Main {

  public static long modpow(long a, long n, long mod) {
    long ret = 1;
    long mul = a;
    for (; n > 0; n >>>= 1) {
      if ((n & 1) == 1) {
        ret = (ret * mul) % mod;
      }
      mul = (mul * mul) % mod;
    }
    return ret;
  }
  
  
  private static void solve() {
    int s = ni();

    long ret = 0;
    int mod = 1_000_000_007;
    
    long total = 1;
    int R = 1;
    int NL = 10, NR = 10, LC = 1, RC = 1;
    for (int L = 1 ; L < 100000000;) {
      while (total < s) {
        R ++;
        if (R == NR) {
          NR *= 10;
          RC ++;
        }
        total += RC;
      }

      if (total == s) {
        ret ++;
      }
      total -= LC;
      L ++;
      if (L == NL) {
        NL *= 10;
        LC ++;
      }
    }
    
    for (int i = 1; i <= s; i ++) {
      int div = s / i;
      if (div <= 8) continue;
      if (s % i > 0) {
        ret ++;
      } else {
        ret += 9 * modpow(10, div - 1, mod) - i + 1 + mod;;
        ret %= mod;
      }
    }
    System.out.println(ret % mod);
  }





  public static void main(String[] args) {
    new Thread(null, new Runnable() {
      @Override
      public void run() {
        long start = System.currentTimeMillis();
        String debug = System.getProperty("debug");
        if (debug != null) {
          try {
            is = java.nio.file.Files.newInputStream(java.nio.file.Paths.get(debug));
          } catch (Exception e) {
            throw new RuntimeException(e);
          }
        }
        reader = new java.io.BufferedReader(new java.io.InputStreamReader(is), 32768);
        solve();
        out.flush();
        tr((System.currentTimeMillis() - start) + "ms");
      }
    }, "", 64000000).start();
  }

  private static java.io.InputStream is = System.in;
  private static java.io.PrintWriter out = new java.io.PrintWriter(System.out);
  private static java.util.StringTokenizer tokenizer = null;
  private static java.io.BufferedReader reader;

  public static String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }

  private static double nd() {
    return Double.parseDouble(next());
  }

  private static long nl() {
    return Long.parseLong(next());
  }

  private static int[] na(int n) {
    int[] a = new int[n];
    for (int i = 0; i < n; i++)
      a[i] = ni();
    return a;
  }

  private static char[] ns() {
    return next().toCharArray();
  }

  private static long[] nal(int n) {
    long[] a = new long[n];
    for (int i = 0; i < n; i++)
      a[i] = nl();
    return a;
  }

  private static int[][] ntable(int n, int m) {
    int[][] table = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[i][j] = ni();
      }
    }
    return table;
  }

  private static int[][] nlist(int n, int m) {
    int[][] table = new int[m][n];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[j][i] = ni();
      }
    }
    return table;
  }

  private static int ni() {
    return Integer.parseInt(next());
  }

  private static void tr(Object... o) {
    if (is != System.in)
      System.out.println(java.util.Arrays.deepToString(o));
  }
}