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 Haskell (GHC 8.8.3) (judge verdict: Accepted)

import Control.Monad
import Data.Maybe
import qualified Data.ByteString.Char8 as BS
import Data.List
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM

readInt = fst . fromJust . BS.readInt
readIntList = map readInt . BS.words
getInt = readInt <$> BS.getLine
getIntList = readIntList <$> BS.getLine

main = do
    n' <- getInt
    as' <- getIntList
    let (o,as) = partition (== 1) as'
    let lo = length o
    let n = length as
    let ps = concatMap factorize as
    let vec = VU.create $ do
            v <- VUM.replicate 1000001 (0 :: Int)
            forM_ ps $ \i -> do
                VUM.modify v succ i
            return v
    let pmax = VU.maximum $ VU.drop 2 vec
    let ans | n == 0 = "pairwise coprime"
            | pmax == 1 = "pairwise coprime"
            | pmax /= n = "setwise coprime"
            | lo /= 0 = "setwise coprime"
            | otherwise = "not coprime"
    putStrLn ans

f n k
  | n < k = n
  | n `mod` k == 0 = f (n `div` k) k
  | otherwise = f (n `mod` k) k
 
primes = map fromIntegral primes'
  where
    primes' = [2, 3, 5] ++ f 5 7 (drop 2 primes')
    f m s (p:ps) = [n | n <- ns, gcd m n == 1] ++ f (m * p) (p * p) ps
      where
        ns = [x + y | x <- [s,s + 6 .. p * p - 2], y <- [0, 4]]

factorize 1 = [1]
factorize n = format $ factorize' n primes
  where
    format ps = map head $ group ps
    factorize' n ps@(p:ps')
      | p * p > n = [n]
      | rem n p == 0 = p : factorize' (div n p) ps
      | otherwise = factorize' n ps'