Problem:
Score : 500 points

Problem Statement

We have N integers. The i-th number is A_i.

\{A_i\} is said to be pairwise coprime when GCD(A_i,A_j)=1 holds for every pair (i, j) such that 1\leq i < j \leq N.

\{A_i\} is said to be setwise coprime when \{A_i\} is not pairwise coprime but GCD(A_1,\ldots,A_N)=1.

Determine if \{A_i\} is pairwise coprime, setwise coprime, or neither.

Here, GCD(\ldots) denotes greatest common divisor.

Constraints

2 \leq N \leq 10^6

1 \leq A_i\leq 10^6

Input

Input is given from Standard Input in the following format:

N
A_1 \ldots A_N

Output

If \{A_i\} is pairwise coprime, print pairwise coprime; if \{A_i\} is setwise coprime, print setwise coprime; if neither, print not coprime.

Sample Input 1

3
3 4 5

Sample Output 1

pairwise coprime

GCD(3,4)=GCD(3,5)=GCD(4,5)=1, so they are pairwise coprime.

Sample Input 2

3
6 10 15

Sample Output 2

setwise coprime

Since GCD(6,10)=2, they are not pairwise coprime. However, since GCD(6,10,15)=1, they are setwise coprime.

Sample Input 3

3
6 10 16

Sample Output 3

not coprime

GCD(6,10,16)=2, so they are neither pairwise coprime nor setwise coprime.

Solution in Rust (1.42.0) (judge verdict: Time Limit Exceeded)

use proconio::{input, fastout};
use std::collections::HashMap;

#[fastout]
fn main() {
    input! {
        n: usize,
        aa: [usize; n],
    }

    let mut ff = vec![];
    let mut is_pairwise = true;
    let mut g = aa[0];
    let max = aa.iter().copied().max().unwrap();
    let primes = sieve(max);

    for &a in aa.iter() {
        g = gcd(g, a);
        ff.push(factors(a, &primes));
    }

    let mut used = vec![false; 1_000_001];
    for f in ff.iter() {
        for (&k, _) in f.iter() {
            if used[k] {
                is_pairwise = false;
            }
            used[k] = true;
        }
    }


    if g != 1 {
        println!("not coprime");
    } else if is_pairwise {
        println!("pairwise coprime");
    } else {
        println!("setwise coprime");
    }
}

fn factors(n: usize, primes: &[usize]) -> HashMap<usize, usize> {
    let mut factors = HashMap::new();
    for &i in primes.iter() {
        if n % i == 0 {
            *factors.entry(i).or_insert(0) += 1;
        }
    }
    if factors.is_empty() && n > 1 {
        *factors.entry(n).or_insert(0) += 1;
    }

    factors
}

fn sieve(n: usize) -> Vec<usize> {
    let mut primes = vec![];
    let mut is_prime = vec![true; n + 1];
    for i in 2..n+1 {
        if !is_prime[i] {
            continue;
        }
        primes.push(i);
        for j in 1..n/i+1 {
            is_prime[i * j] = false;
        }
    }
    primes
}

fn gcd(a: usize, b: usize) -> usize {
    if a % b == 0 {
        b
    } else {
        gcd(b, a % b)
    }
}