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: Accepted)

#![allow(unused_macros)]
#![allow(dead_code)]
#![allow(unused_imports)]

use itertools::Itertools;
use num_integer::gcd;
use proconio::*;
use text_io::*;
use std::collections::HashSet;

const U_INF: usize = 1 << 60;
const I_INF: isize = 1 << 60;

fn main() {
    input! {
        n:usize,
        a:[usize;n],
    }

    let mut sum = 0;
    for &ai in &a {
        sum = gcd(sum, ai);
    }

    let max = a.iter().max().unwrap();

    let pf = prime_factorization(*max);
    let mut primes = vec![0; max + 10];

    for ai in a {
        for &p in &pf[ai] {
            primes[p] += 1;
        }
    }

    let pairwise = *primes.iter().max().unwrap() <= 1;
    let setwise = sum == 1;

    let ans = if pairwise {
        "pairwise coprime"
    } else if setwise {
        "setwise coprime"
    } else {
        "not coprime"
    };
    
    println!("{}", ans);
}

fn prime_factorization(n: usize) -> Vec<HashSet<usize>> {
    let mut v = vec![HashSet::new(); n + 10];

    for i in 2..=n {
        if v[i].len() <= 1 {
            v[i].insert(i);
            let mut j = 2;
            while i * j <= n {
                v[i * j].insert(i);
                j += 1;
            }
        }
    }

    v
}