Problem:
Score : 400 points

Problem Statement

Given is a positive integer N. Consider repeatedly applying the operation below on N:

First, choose a positive integer z satisfying all of the conditions below:

z can be represented as z=p^e, where p is a prime number and e is a positive integer;

z divides N;

z is different from all integers chosen in previous operations.

Then, replace N with N/z.

Find the maximum number of times the operation can be applied.

Constraints

All values in input are integers.

1 \leq N \leq 10^{12}

Input

Input is given from Standard Input in the following format:

N

Output

Print the maximum number of times the operation can be applied.

Sample Input 1

24

Sample Output 1

3

We can apply the operation three times by, for example, making the following choices:

Choose z=2 (=2^1). (Now we have N=12.)

Choose z=3 (=3^1). (Now we have N=4.)

Choose z=4 (=2^2). (Now we have N=1.)

Sample Input 2

1

Sample Output 2

0

We cannot apply the operation at all.

Sample Input 3

64

Sample Output 3

3

We can apply the operation three times by, for example, making the following choices:

Choose z=2 (=2^1). (Now we have N=32.)

Choose z=4 (=2^2). (Now we have N=8.)

Choose z=8 (=2^3). (Now we have N=1.)

Sample Input 4

1000000007

Sample Output 4

1

We can apply the operation once by, for example, making the following choice:

z=1000000007 (=1000000007^1). (Now we have N=1.)

Sample Input 5

997764507000

Sample Output 5

7

Solution in Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.util.Arrays;
import java.util.Random;

public class Main {

  private static void solve() {
    long n = nl();
    int[] primes = sieveEratosthenes(2000000);

    long ret = 0;
    for (long[] v : factorX(n, primes)) {
      long x = v[1];
      long k = 0;
      while (x >= k + 1) {
        k++;
        x -= k;
      }
      ret += k;
    }
    System.out.println(ret);
  }

  static long[][] factorX(long n, int[] primes) {
    long[][] ret = new long[20][2];
    int rp = 0;
    for (int p : primes) {
      if ((long) p * p > n)
        break;
      int i;
      for (i = 0; n % p == 0; n /= p, i++)
        ;
      if (i > 0) {
        ret[rp][0] = p;
        ret[rp][1] = i;
        rp++;
      }
    }
    if (n == 1)
      return Arrays.copyOf(ret, rp);

    // P^2
    long sq = (long) Math.sqrt(n);
    for (long u = Math.max(2, sq - 2); u <= sq + 2; u++) {
      if (u * u == n) {
        ret[rp][0] = u;
        ret[rp][1] = 2;
        rp++;
        return Arrays.copyOf(ret, rp);
      }
    }

    // Prime
    if (doMirrorRabin(n)) {
      ret[rp][0] = n;
      ret[rp][1] = 1;
      rp++;
      return Arrays.copyOf(ret, rp);
    }

    // P*Q
    long f = rho(n);
    if (f == -1)
      throw new ArithmeticException();
    if (f > n / f)
      f = n / f;
    ret[rp][0] = f;
    ret[rp][1] = 1;
    rp++;
    ret[rp][0] = n / f;
    ret[rp][1] = 1;
    rp++;
    return Arrays.copyOf(ret, rp);
  }

  public static long mul(long a, long b, long mod) {
    a %= mod;
    long ret = 0;
    int x = 63 - Long.numberOfLeadingZeros(b);
    for (; x >= 0; x--) {
      ret += ret;
      if (ret >= mod)
        ret -= mod;
      if (b << ~x < 0) {
        ret += a;
        if (ret >= mod)
          ret -= mod;
      }
    }
    return ret;
  }

  public static boolean doMirrorRabin(long n) {
    // int[] P = {2, 7, 61}; // n<4759123141
    int[] P = { 2, 3, 5, 7, 11, 13, 17, 19, 23 }; // n=long
    int s = Long.numberOfTrailingZeros(n - 1);
    long d = n - 1 >> s;
    outer: for (int a : P) {
      if (a >= n)
        continue;

      long mul = a;
      long ad = 1;
      for (long e = d; e > 0; e >>>= 1) {
        if ((e & 1) == 1)
          ad = mul(ad, mul, n);
        mul = mul(mul, mul, n);
      }
      if (ad == 1)
        continue;

      for (int r = 0; r < s; r++) {
        if (ad == n - 1)
          continue outer;
        ad = mul(ad, ad, n);
      }
      return false;
    }
    return true;
  }

  static long rho(long n) {
    Random gen = new Random();
    for (int u = 0; u < 100; u++) {
      long ran = (gen.nextLong() & Long.MAX_VALUE) % n;
      long x = 2, y = 2, d = 1;
      while (d == 1) {
        x = (mul(x, x, n) + ran) % n;
        y = (mul(y, y, n) + ran) % n;
        y = (mul(y, y, n) + ran) % n;
        d = gcd(Math.abs(x - y), n);
      }
      if (d < n)
        return d;
    }
    return -1;
  }

  public static long gcd(long a, long b) {
    while (b > 0) {
      long c = a;
      a = b;
      b = c % b;
    }
    return a;
  }

  public static int[] sieveEratosthenes(int n) {
    if (n <= 32) {
      int[] primes = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 };
      for (int i = 0; i < primes.length; i++) {
        if (n < primes[i]) {
          return Arrays.copyOf(primes, i);
        }
      }
      return primes;
    }

    int u = n + 32;
    double lu = Math.log(u);
    int[] ret = new int[(int) (u / lu + u / lu / lu * 1.5)];
    ret[0] = 2;
    int pos = 1;

    int[] isnp = new int[(n + 1) / 32 / 2 + 1];
    int sup = (n + 1) / 32 / 2 + 1;

    int[] tprimes = { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 };
    for (int tp : tprimes) {
      ret[pos++] = tp;
      int[] ptn = new int[tp];
      for (int i = (tp - 3) / 2; i < tp << 5; i += tp)
        ptn[i >> 5] |= 1 << i;
      for (int j = 0; j < sup; j += tp) {
        for (int i = 0; i < tp && i + j < sup; i++) {
          isnp[j + i] |= ptn[i];
        }
      }
    }

    // 3,5,7
    // 2x+3=n
    int[] magic = { 0, 1, 23, 2, 29, 24, 19, 3, 30, 27, 25, 11, 20, 8, 4, 13, 31, 22, 28, 18, 26, 10, 7, 12, 21, 17, 9,
        6, 16, 5, 15, 14 };
    int h = n / 2;
    for (int i = 0; i < sup; i++) {
      for (int j = ~isnp[i]; j != 0; j &= j - 1) {
        int pp = i << 5 | magic[(j & -j) * 0x076be629 >>> 27];
        int p = 2 * pp + 3;
        if (p > n)
          break;
        ret[pos++] = p;
        if ((long) p * p > n)
          continue;
        for (int q = (p * p - 3) / 2; q <= h; q += p)
          isnp[q >> 5] |= 1 << q;
      }
    }

    return Arrays.copyOf(ret, pos);
  }

  public static void main(String[] args) {
    new Thread(null, new Runnable() {
      @Override
      public void run() {
        long start = System.currentTimeMillis();
        String debug = args.length > 0 ? args[0] : null;
        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));
  }
}