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 Rust (1.42.0) (judge verdict: Accepted)

//! https://github.com/hatoo/competitive-rust-snippets
//!
//! MIT License
//!
//! Copyright (c) 2018 hatoo
//!
//! Permission is hereby granted, free of charge, to any person obtaining a copy
//! of this software and associated documentation files (the "Software"), to deal
//! in the Software without restriction, including without limitation the rights
//! to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
//! copies of the Software, and to permit persons to whom the Software is
//! furnished to do so, subject to the following conditions:
//!
//! The above copyright notice and this permission notice shall be included in all
//! copies or substantial portions of the Software.
//!
//! THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
//! IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
//! FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
//! AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
//! LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
//! OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
//! SOFTWARE.
#![allow(
    unused_imports,
    unused_attributes,
    unused_macros,
    dead_code,
    non_snake_case
)]
use std::cmp::{max, min, Ordering};
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
use std::io::{stdin, stdout, BufWriter, Write};
use std::iter::FromIterator;
#[macro_export]
macro_rules !get {(@inner [$src :expr ] chars ) =>{{let mut buf =String ::new () ;$src .read_line (&mut buf ) .unwrap () ;buf .trim () .chars () .collect ::<Vec <char >>() } } ;(@inner [$src :expr ] usize1 ) =>{{get !(@inner [$src ] usize ) -1 } } ;(@inner [$src :expr ] [usize1 ] ) =>{{get !(@inner [$src ] [usize ] ) .into_iter () .map (|v |v -1 ) .collect ::<Vec <usize >>() } } ;(@inner [$src :expr ] [[usize1 ] ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] [usize1 ] ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [usize1 ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] [usize1 ] ) ) .flatten () .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [[chars ] ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] chars ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [chars ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] chars ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [($($tt :tt ) ,*) ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] ($($tt ) ,*) ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] ($($tt :tt ) ,*) ) =>{{let mut buf :String =String ::new () ;$src .read_line (&mut buf ) .unwrap () ;let mut iter =buf .split_whitespace () ;($(get !(@inner_elem_parse [$tt ] iter .next () .unwrap () ) ,) *) } } ;(@inner [$src :expr ] [$t :ty ] ) =>{{let mut buf =String ::new () ;$src .read_line (&mut buf ) .unwrap () ;buf .trim () .split_whitespace () .map (|t |t .parse ::<$t >() .unwrap () ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [[$t :ty ] ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] [$t ] ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [$t :ty ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] [$t ] ) ) .flatten () .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] $t :ty ) =>{{let mut buf =String ::new () ;$src .read_line (&mut buf ) .unwrap () ;buf .trim () .split_whitespace () .next () .unwrap () .parse ::<$t >() .unwrap () } } ;(@inner_elem_parse [usize1 ] $elem :expr ) =>{{get !(@inner_elem_parse [usize ] $elem ) -1 } } ;(@inner_elem_parse [$t :ty ] $elem :expr ) =>{{$elem .parse ::<$t >() .unwrap () } } ;($tt :tt ) =>{{use std ::io ::BufRead ;let get_stdin =std ::io ::stdin () ;let mut locked_stdin =get_stdin .lock () ;get !(@inner [&mut locked_stdin ] $tt ) } } ;}
macro_rules !debug {($($a :expr ) ,*$(,) *) =>{#[cfg (debug_assertions ) ] eprintln !(concat !($("| " ,stringify !($a ) ,"={:?} " ) ,*,"|" ) ,$(&$a ) ,*) ;} ;}
macro_rules !echo {($($a :expr ) ,*) =>{let mut s =Vec ::new () ;$(s .push (format !("{}" ,$a ) ) ;) *println !("{}" ,s .join (" " ) ) ;} }
#[macro_export]
macro_rules !chmin {($base :expr ,$($cmps :expr ) ,+$(,) *) =>{{let cmp_min =min !($($cmps ) ,+) ;if $base >cmp_min {$base =cmp_min ;true } else {false } } } ;}
#[macro_export]
macro_rules !chmax {($base :expr ,$($cmps :expr ) ,+$(,) *) =>{{let cmp_max =max !($($cmps ) ,+) ;if $base <cmp_max {$base =cmp_max ;true } else {false } } } ;}
#[macro_export]
macro_rules !min {($a :expr $(,) *) =>{{$a } } ;($a :expr ,$b :expr $(,) *) =>{{std ::cmp ::min ($a ,$b ) } } ;($a :expr ,$($rest :expr ) ,+$(,) *) =>{{std ::cmp ::min ($a ,min !($($rest ) ,+) ) } } ;}
#[macro_export]
macro_rules !max {($a :expr $(,) *) =>{{$a } } ;($a :expr ,$b :expr $(,) *) =>{{std ::cmp ::max ($a ,$b ) } } ;($a :expr ,$($rest :expr ) ,+$(,) *) =>{{std ::cmp ::max ($a ,max !($($rest ) ,+) ) } } ;}
const BIG_STACK_SIZE: bool = false;
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}
fn solve() {
    let N = get!(u128);
    let fac = prime_fac(N);
    debug!(fac);
    let mut ans = 0;
    for (_, cnt) in fac {
        ans += calc(cnt);
    }
    echo!(ans);
}

fn calc(e: u128) -> u128 {
    let mut ok = 0u128;
    let mut ng = 1_000_000_000u128;
    while ng - ok > 1 {
        let mid = (ok + ng) / 2;
        if mid * (mid + 1) <= 2 * e {
            ok = mid;
        } else {
            ng = mid;
        }
    }
    ok
}

pub fn prime_fac(n: u128) -> Vec<(u128, u128)> {
    let mut n = n;
    let mut ret = Vec::new();
    let mut cur = 2;
    loop {
        if cur * cur > n {
            break;
        }
        if n % cur != 0 {
            cur += 1;
            continue;
        }
        let mut ex = 0;
        while n % cur == 0 {
            ex += 1;
            n /= cur;
        }
        if ex > 0 {
            ret.push((cur, ex));
        }
        cur += 1;
    }
    if n != 1 {
        ret.push((n, 1));
    }
    ret
}

pub trait Int:
    std::ops::Add<Output = Self>
    + std::ops::Sub<Output = Self>
    + std::ops::Mul<Output = Self>
    + std::ops::Div<Output = Self>
    + std::ops::Rem<Output = Self>
    + std::hash::Hash
    + PartialEq
    + Eq
    + PartialOrd
    + Ord
    + Copy
{
    fn zero() -> Self;
    fn one() -> Self;
    fn next(self) -> Self;
    fn prev(self) -> Self;
    fn sqrt_floor(self) -> Self {
        if self < Self::zero() {
            return Self::zero();
        }
        let two = Self::one().next();
        let mut ok = Self::zero();
        let mut ng = self.next();
        while ng - ok > Self::one() {
            let mid = (ng + ok) / two;
            if mid * mid <= self {
                ok = mid;
            } else {
                ng = mid;
            }
        }
        ok
    }
    fn is_zero(&self) -> bool {
        *self == Self::zero()
    }
    fn is_one(&self) -> bool {
        *self == Self::one()
    }
    fn div_rem(&self, other: &Self) -> (Self, Self) {
        ((*self) / (*other), (*self) % (*other))
    }
    fn div_floor(&self, other: &Self) -> Self {
        let (d, r) = self.div_rem(other);
        if (r > Self::zero() && *other < Self::zero())
            || (r < Self::zero() && *other > Self::zero())
        {
            d.prev()
        } else {
            d
        }
    }
    fn mod_floor(&self, other: &Self) -> Self {
        let r = *self % *other;
        if (r > Self::zero() && *other < Self::zero())
            || (r < Self::zero() && *other > Self::zero())
        {
            r + *other
        } else {
            r
        }
    }
}
macro_rules !impl_int_for_numerics {($($t :ty ) *) =>{$(impl Int for $t {fn zero () ->Self {0 } fn one () ->Self {1 } fn next (self ) ->Self {self +Self ::one () } fn prev (self ) ->Self {self -Self ::one () } } ) *} }
impl_int_for_numerics !(u8 i8 u16 i16 u32 i32 u64 i64 usize isize u128);
pub trait Prime<T: Int> {
    fn lower_primes(&self) -> Vec<T>;
    fn factorize(&self) -> std::collections::HashMap<T, u128>;
}

impl<T> Prime<T> for T
where
    T: Int,
{
    /// エラトステネスの篩を用いてself以下の素数を求める
    /// 計算量: O(n log log n)
    fn lower_primes(&self) -> Vec<T> {
        let &this = self;
        let mut v = Vec::new();
        if this <= T::one() {
            return v;
        }
        let mut deque = std::collections::VecDeque::new();
        let mut t = T::one().next();
        while t <= this {
            deque.push_back(t);
            t = t.next();
        }
        let mut p = match deque.pop_front() {
            Some(x) => x,
            None => return v,
        };
        v.push(p);
        while p * p <= this {
            deque = deque
                .iter()
                .filter(|&&x| x % p != T::zero())
                .copied()
                .collect();
            p = match deque.pop_front() {
                Some(x) => x,
                None => return v,
            };
            v.push(p);
        }
        for n in deque {
            v.push(n);
        }
        v
    }
    /// エラトステネスの篩を用いてselfを素因数分解する
    fn factorize(&self) -> std::collections::HashMap<T, u128> {
        let mut ret = std::collections::HashMap::new();
        let primes = self.sqrt_floor().lower_primes();
        let mut tmp = *self;
        for prime in primes {
            while tmp % prime == T::zero() {
                tmp = tmp / prime;
                *ret.entry(prime).or_insert(0) += 1;
            }
        }
        if tmp > T::one() {
            *ret.entry(tmp).or_insert(0) += 1;
        }
        ret
    }
}