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 C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;

class Program {

    static void Main (string[] args) {
        long N = long.Parse (Console.ReadLine ());
        var factor = new Factorization (N);
        var dic = new Dictionary<long, long> ();
        foreach (var value in factor.Calc ()) {
            if (dic.ContainsKey (value)) {
                dic[value]++;
            } else {
                dic.Add (value, 1);
            }
        }
        long ans = 0;
        foreach ((long key, long value) in dic) {
            ans += (long) Math.Floor ((-1 + Math.Sqrt (1 + 8 * (double) value)) / 2);
        }
        Console.WriteLine (ans);
    }

    //1000000007
}

public class Factorization {
    private static long N;

    public Factorization (long n) { N = n; }

    public static long GCD (long a, long b) {
        if (a < b) return GCD (b, a);
        long r;
        while (b != 0) {
            r = a % b;
            a = b;
            b = r;
        }
        return a;
    }

    public static bool IsPrime (long n) {
        if (n < 2) return false;
        if (n == 2) return true;
        if (n % 2 == 0) return false;

        double root = Math.Sqrt (n);
        for (int i = 3; i <= root; i += 2) {
            if (n % i == 0) return false;
        }
        return true;
    }

    private static int[] TryPrimes = new int[] { 2, 3, 5, 7, 11 };
    public long TryAllocation (long p) {
        long count = 0;
        while (N % p == 0) {
            count++;
            N /= p;
        }
        return count;
    }

    private int[] xi = new int[] { 13, 17, 19, 23, 29 };
    private long f (long x, int suffix) {
        return (xi[suffix % xi.Length] * x + suffix) % N;
    }

    public long PollardsRho (long n, int suffix = 0) {
        if (IsPrime (n)) return n;
        long x = 2, y = 2, d = 1;
        while (d == 1) {
            x = f (x, suffix);
            y = f (f (y, suffix), suffix);
            d = GCD (Math.Abs (x - y), n);
        }
        if (d == n) {
            return PollardsRho (n, suffix + 1);
        }
        return d;
    }

    public IEnumerable<long> Calc () {
        foreach (var prime in TryPrimes) {
            var box = TryAllocation (prime);
            for (int i = 0; i < box; i++) yield return prime;
        }

        while (true) {
            if (N == 1) break;
            var ret = PollardsRho (N);
            yield return ret;
            N /= ret;
        }
    }
}
//dotnet run --project test