Problem:
Score : 1100 points

Problem Statement

Snuke got positive integers s_1,...,s_N from his mother, as a birthday present. There may be duplicate elements.

He will circle some of these N integers. Since he dislikes cubic numbers, he wants to ensure that if both s_i and s_j (i ≠ j) are circled, the product s_is_j is not cubic. For example, when s_1=1,s_2=1,s_3=2,s_4=4, it is not possible to circle both s_1 and s_2 at the same time. It is not possible to circle both s_3 and s_4 at the same time, either.

Find the maximum number of integers that Snuke can circle.

Constraints

1 ≦ N ≦ 10^5

1 ≦ s_i ≦ 10^{10}

All input values are integers.

Input

The input is given from Standard Input in the following format:

N
s_1
:
s_N

Output

Print the maximum number of integers that Snuke can circle.

Sample Input 1

8
1
2
3
4
5
6
7
8

Sample Output 1

6

Snuke can circle 1,2,3,5,6,7.

Sample Input 2

6
2
4
8
16
32
64

Sample Output 2

3

Sample Input 3

10
1
10
100
1000000007
10000000000
1000000009
999999999
999
999
999

Sample Output 3

9

Solution in Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports, unused_macros)]

use kyoproio::*;
use std::{
    collections::*,
    io::{self, prelude::*},
    iter,
    mem::{replace, swap},
};

fn main() -> io::Result<()> {
    std::thread::Builder::new()
        .stack_size(64 * 1024 * 1024)
        .spawn(|| {
            let stdin = io::stdin();
            let stdout = io::stdout();
            run(KInput::new(stdin.lock()), io::BufWriter::new(stdout.lock()))
        })?
        .join()
        .unwrap();
    Ok(())
}

fn run<I: Input, O: Write>(mut kin: I, mut out: O) {
    macro_rules! output { ($($args:expr),+) => { write!(&mut out, $($args),+).unwrap(); }; }
    macro_rules! outputln {
        ($($args:expr),+) => { output!($($args),+); outputln!(); };
        () => { output!("\n"); if cfg!(debug_assertions) { out.flush().unwrap(); } }
    }

    let n: usize = kin.input();
    let s: Vec<usize> = kin.seq(n);
    
    let ps = primes(100000);
    let mut pcs = Vec::new();
    for &p in &ps {
        let c = p * p * p;
        if c > 10000000000 {
            break;
        }
        pcs.push(c);
    }
    let mut map = BTreeMap::new();
    for &x in &s {
        let mut x = x;
        for &c in &pcs {
            if c > x {
                break;
            }
            while x % c == 0 {
                x /= c;
            }
        }
        *map.entry(x).or_insert(0) += 1;
    }

    let mut ans = 0;
    let mut used = BTreeSet::new();
    for (&x, &c) in map.iter() {
        if x == 1 {
            ans += 1;
            continue;
        }
        if used.contains(&x) {
            continue;
        }
        let mut x = x;
        let mut y = 1usize;
        for &p in &ps {
            if p * p > x {
                break;
            }
            let mut k = 0;
            while x % p == 0 {
                x /= p;
                k += 1;
            }
            if k == 1 {
                y = y.saturating_mul(p.saturating_mul(p));
            }
            if k == 2 {
                y = y.saturating_mul(p);
            }
        }
        y = y.saturating_mul(x.saturating_mul(x));

        if let Some(&d) = map.get(&y) {
            used.insert(y);
            ans += c.max(d);
        } else {
            ans += c;
        }
    }

    outputln!("{}", ans);

    // eprintln!("{:?}", map);
}

pub fn primes(n: usize) -> Vec<usize> {
    // 1, 7, 11, 13, 17, 19, 23, 29
    const SKIP: [u8; 8] = [6, 4, 2, 4, 2, 4, 6, 2];
    const XTOI: [u8; 15] = [
        0, 0, 0, 1, 0, 2, 3, 0, 4, 5, 0, 6, 0, 0, 7,
    ];
    let mut sieve = vec![0u8; n / 30 + 1];
    let mut ps = vec![2, 3, 5];
    if n <= 4 {
        ps.truncate([0, 0, 1, 2, 2][n]);
        return ps;
    }
    let mut x = 7;
    let mut i = 1;
    while x <= n {
        if sieve[i / 8] & 1 << i % 8 == 0 {
            ps.push(x);
            let mut j = i;
            let mut y = x * x;
            while y <= n {
                sieve[y / 30] |= 1 << XTOI[y / 2 % 15];
                y += x * SKIP[j % 8] as usize;
                j += 1;
            }
        }
        x += SKIP[i % 8] as usize;
        i += 1;
    }
    ps
}

// -----------------------------------------------------------------------------
pub mod kyoproio {
    use std::io::prelude::*;
    pub trait Input {
        fn str(&mut self) -> &str;
        fn input<T: InputParse>(&mut self) -> T {
            T::input(self)
        }
        fn iter<T: InputParse>(&mut self) -> Iter<T, Self> {
            Iter(self, std::marker::PhantomData)
        }
        fn seq<T: InputParse, B: std::iter::FromIterator<T>>(&mut self, n: usize) -> B {
            self.iter().take(n).collect()
        }
    }
    pub struct KInput<R> {
        src: R,
        buf: String,
        pos: usize,
    }
    impl<R: BufRead> KInput<R> {
        pub fn new(src: R) -> Self {
            Self {
                src,
                buf: String::with_capacity(1024),
                pos: 0,
            }
        }
    }
    impl<R: BufRead> Input for KInput<R> {
        fn str(&mut self) -> &str {
            loop {
                if self.pos >= self.buf.len() {
                    self.pos = 0;
                    self.buf.clear();
                    if self.src.read_line(&mut self.buf).expect("io error") == 0 {
                        return &self.buf;
                    }
                }
                let range = self.pos
                    ..self.buf[self.pos..]
                        .find(|c: char| c.is_ascii_whitespace())
                        .map(|i| i + self.pos)
                        .unwrap_or_else(|| self.buf.len());
                self.pos = range.end + 1;
                if range.end > range.start {
                    return &self.buf[range];
                }
            }
        }
    }
    pub struct Iter<'a, T, I: ?Sized>(&'a mut I, std::marker::PhantomData<*const T>);
    impl<'a, T: InputParse, I: Input + ?Sized> Iterator for Iter<'a, T, I> {
        type Item = T;
        fn next(&mut self) -> Option<T> {
            Some(self.0.input())
        }
    }
    pub trait InputParse: Sized {
        fn input<I: Input + ?Sized>(src: &mut I) -> Self;
    }
    impl InputParse for Vec<u8> {
        fn input<I: Input + ?Sized>(src: &mut I) -> Self {
            src.str().as_bytes().to_owned()
        }
    }
    macro_rules! from_str_impl {
        { $($T:ty)* } => {
            $(impl InputParse for $T {
                fn input<I: Input + ?Sized>(src: &mut I) -> Self {
                    src.str().parse::<$T>().expect("parse error")
                }
            })*
        }
    }
    from_str_impl! {
        String char bool f32 f64 isize i8 i16 i32 i64 i128 usize u8 u16 u32 u64 u128
    }
    macro_rules! tuple_impl {
        ($H:ident $($T:ident)*) => {
            impl<$H: InputParse, $($T: InputParse),*> InputParse for ($H, $($T),*) {
                fn input<I: Input + ?Sized>(src: &mut I) -> Self {
                    ($H::input(src), $($T::input(src)),*)
                }
            }
            tuple_impl!($($T)*);
        };
        () => {}
    }
    tuple_impl!(A B C D E F G);
    #[macro_export]
    macro_rules! kdbg {
        ($($v:expr),*) => {
            if cfg!(debug_assertions) { dbg!($($v),*) } else { ($($v),*) }
        }
    }
}