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: Time Limit Exceeded)

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

using E = System.Linq.Enumerable;

public class Program
{
    public static void Main()
    {
        var n = long.Parse(Console.ReadLine());

        List<int> factorExponents = PrimeFactorize(n);

        var count = 0;
        for (int i = 0; i < factorExponents.Count; ++i)
        {
            var (ok, ng) = (1, factorExponents[i] + 1);
            while (Math.Abs(ok - ng) > 1)
            {
                var mid = (ng - ok) / 2 + ok;
                if (mid * (mid + 1) / 2 <= factorExponents[i])
                {
                    ok = mid;
                }
                else
                {
                    ng = mid;
                }
            }

            count += ok;
            factorExponents[i] -= ok * (ok + 1) / 2;
        }

        Console.WriteLine(count);
    }

    private static List<int> PrimeFactorize(long n)
    {
        var factorExponents = new List<int>();
        for (var p = 2; p * p <= n; p += 2 - (1 - p % 2))
        {
            var e = 0;
            var t = n / p;
            while (t * p == n)
            {
                (n, t) = (t, t / p);
                ++e;
            }

            if (e > 0)
            {
                factorExponents.Add(e);
            }
        }
        if (n != 1)
        {
            factorExponents.Add(1);
        }

        return factorExponents;
    }
}