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)

{-# LANGUAGE MultiWayIf #-}

module Main where

import Control.Monad
import Control.Monad.Extra
import qualified Data.Array as A
import qualified Data.Array.IO as AIO
import qualified Data.Array.Unboxed as AU
--import qualified Data.Heap                     as H

import Data.Bits
import qualified Data.ByteString as BS
import qualified Data.ByteString.Char8 as BSC8
import Data.Char
import qualified Data.Graph.Inductive.Graph as G
import qualified Data.Graph.Inductive.PatriciaTree as G
import Data.Int
import Data.List
import qualified Data.Map.Strict as M
import Data.Maybe
import Data.Monoid
import Data.Ord
import qualified Data.Sequence as SQ
import qualified Data.Set as S
import qualified Data.Vector as V
import Data.Vector.Algorithms.Search
import qualified Data.Vector.Mutable as VM
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import Debug.Trace
import Prelude

main =
  do
    --aN       <- getI
    --[aA, aB] <- getIL
    --aS       <- getLine

    aN <- getI
    aA <- getIL

    aaa <- era
    bbb@(ddd, eee, _) <-
      foldM
        ( \acc@(flg, ccc, hist) x ->
            if flg == True
              then do
                let primes1 = primes aaa x
                if histMember primes1 hist
                  then return (False, gcd ccc x, hist)
                  else return (True, gcd ccc x, histInsert primes1 hist)
              else return (False, gcd ccc x, hist)
        )
        (True, head aA, histInsert (primes aaa (head aA)) S.empty)
        (tail aA)

    if ddd == True
      then putStrLn "pairwise coprime"
      else
        if eee == 1
          then putStrLn "setwise coprime"
          else putStrLn "not coprime"
    return ()

histInsert primes hist =
  foldl ((flip S.insert)) hist primes

histMember [] hist = False
histMember (prime : primes) hist =
  if S.member prime hist
    then True
    else histMember primes hist

primes aaa x
  | x == 0 || x == 1 = []
  | otherwise =
    let (prime, quo) = primesP aaa x
     in if quo == 1
          then [prime]
          else prime : (primes aaa quo)

primesP aaa x =
  let target = aaa VU.! x
   in if target == 0
        then (x, 1)
        else (target, div x target)

era = do
  aaa <- VUM.replicate (10 ^ 6 + 1) (0 :: Int) :: IO (VUM.IOVector Int)
  eraP aaa 2
  mapM_ (\y -> eraP aaa y) [x * 2 + 1 | x <- [1 .. 500000]]
  VU.freeze aaa

eraP aaa x
  | x > 1000000 = return ()
  | otherwise = modi aaa x 1

modi aaa x n
  | x * n > 1000000 = return ()
  | otherwise = do
    target <- VUM.read aaa (x * n)
    if target == 0
      then do
        VUM.write aaa (x * n) x
        modi aaa x (n + 1)
      else modi aaa x (n + 1)

{-
hMaxPoint :: Int
hMaxPoint = 101
hMaximum :: H.Heap Int -> Int
hMaximum h = hMaxPoint - (H.minimum h)
hInsertMax :: Int -> H.Heap Int -> H.Heap Int
hInsertMax x = H.insert (hMaxPoint - x)
hDeleteMax :: H.Heap Int -> H.Heap Int
hDeleteMax = H.deleteMin
-}
{-}
aroundSquare y x h w =
  [ aYX
    | aYX@(aY, aX) <- [(y - 1, x), (y, x - 1), (y + 1, x), (y, x + 1), (y, x)],
      aY >= 0,
      aY <= (h - 1),
      aX >= 0,
      aX <= (w - 1)
  ]
-}

aroundSquare y x h w =
  [ aYX
    | aYX@(aY, aX) <- [(y + a, x + b) | a <- [-1 .. 1], b <- [-1 .. 1], abs (a) + abs (b) == 1],
      aY >= 1,
      aY <= (h),
      aX >= 1,
      aX <= (w),
      (aY /= y) || (aX /= x)
  ]

warpAroundSquare y x h w =
  [ aYX
    | aYX@(aY, aX) <- [(y + a, x + b) | a <- [-2 .. 2], b <- [-2 .. 2], abs (a) + abs (b) > 1],
      aY >= 1,
      aY <= (h),
      aX >= 1,
      aX <= (w),
      (aY /= y) || (aX /= x)
  ]

splitReadBSC8 :: BSC8.ByteString -> [Int]
splitReadBSC8 bs =
  let (bs1, newBs) = BSC8.splitAt 1 bs
   in if newBs == BSC8.empty
        then [test $ BSC8.readInt bs1]
        else (test $ BSC8.readInt bs1) : (splitReadBSC8 newBs)
  where
    test (Just (i, _)) = i

read2DimChar :: VM.IOVector (VUM.IOVector Char) -> Int -> Int -> IO Char
read2DimChar vec y x = do
  vecY <- VM.read vec y
  VUM.read vecY x

write2DimChar :: VM.IOVector (VUM.IOVector Char) -> Int -> Int -> Char -> IO ()
write2DimChar vec y x t = do
  vecY <- VM.read vec y
  VUM.write vecY x t

readTwoDimVec :: VM.IOVector (VUM.IOVector Int) -> Int -> Int -> IO Int
readTwoDimVec vec y x = do
  vecY <- VM.read vec y
  VUM.read vecY x

writeTwoDimVec :: VM.IOVector (VUM.IOVector Int) -> Int -> Int -> Int -> IO ()
writeTwoDimVec vec y x t = do
  vecY <- VM.read vec y
  VUM.write vecY x t

getVUI :: IO (VU.Vector Int)
getVUI = VU.unfoldr (BSC8.readInt . BSC8.dropWhile isSpace) <$> BSC8.getLine

getRepVUI :: Int -> IO (VU.Vector Int)
getRepVUI n =
  VU.unfoldr (BSC8.readInt . BSC8.dropWhile isSpace)
    . BSC8.unlines
    <$> replicateM n BSC8.getLine

getVS :: IO (V.Vector Char)
getVS = V.fromList <$> getLine

getVUC :: IO (VU.Vector Char)
getVUC = VU.fromList <$> getLine

readI :: BSC8.ByteString -> Int
readI = fst . fromJust . BSC8.readInt

readIL :: BSC8.ByteString -> [Int]
readIL = map readI . BSC8.words

readIT2 :: BSC8.ByteString -> (Int, Int)
readIT2 bs =
  let words = BSC8.words bs in (readI $ head words, readI $ last words)

getI :: IO Int
getI = readI <$> BSC8.getLine

getIL :: IO [Int]
getIL = readIL <$> BSC8.getLine

getIT2 :: IO (Int, Int)
getIT2 = readIT2 <$> BSC8.getLine

getIT2Order :: IO (Int, Int)
getIT2Order =
  getIL
    >>= \x@[xz, xo] -> if xz > xo then return (xo, xz) else return (xz, xo)

getNIS :: Int -> IO [(Int, BSC8.ByteString)]
getNIS 0 = return []
getNIS n = do
  [aFst, aSnd] <- BSC8.words <$> BSC8.getLine
  next <- getNIS (n - 1)
  return ((readI aFst, aSnd) : next)

il2Words :: [Int] -> String
il2Words = unwords . map show

il2Lines :: [Int] -> String
il2Lines = unlines . map show

p :: Int -> Int
p x = x - 1

--modNum = 10 ^ 9 + 7
--modulus = 1000000007
modulus = 1000000007

newtype MInt = M Int

instance Num MInt where
  (M a) + (M b) = M $ mod (a + b) modulus

  (M a) - (M b) = M $ mod (a - b) modulus

  (M a) * (M b) = M $ mod (a * b) modulus

  fromInteger a = M $ mod (fromInteger a) modulus

  abs (M a) = undefined

  signum (M a) = undefined

instance Show MInt where
  show (M a) = show (mod a modulus)

int2MInt :: Int -> MInt
int2MInt a = M (mod a modulus)

mInt2Int :: MInt -> Int
mInt2Int (M a) = a

{-
factsMod :: V.Vector MInt
factsMod = V.scanl (\acc x -> M x * acc) 1 $ V.generate 2000000 (+ 1)

factMod :: Int -> MInt
factMod = (factsMod V.!)
-}

factMod :: Int -> MInt
factMod n
  | n == 0 = 1
  | n == 1 = 1
  | otherwise = M n * (factMod (n - 1))

factDivMod :: Int -> Int -> MInt
factDivMod n k
  | n == k = M 1
  | otherwise = M n * (factDivMod (n - 1) k)

powMod :: MInt -> MInt -> MInt
powMod n mK@(M k)
  | k == 0 = M 1
  | k == 1 = n
  | mod k 2 == 0 = powMod (n ^ 2) (M (div k 2))
  | otherwise = n * powMod (n ^ 2) (M (div k 2))

invMod :: MInt -> MInt
invMod (M 0) = error "inverse of 0"
invMod x = powMod x (M (modulus - 2))

{-
combiMod :: Int -> Int -> MInt
combiMod n k
  | n < 0 || k < 0 || k > n = 0
  | otherwise = factMod n * invMod (factMod (n - k)) * invMod (factMod k)
-}
combiMod :: Int -> Int -> MInt
combiMod n k
  | n < 0 || k < 0 || k > n = 0
  | k > n - k = factDivMod n k * invMod (factMod (n - k))
  | otherwise = factDivMod n (n - k) * invMod (factMod k)

combi :: Int -> Int -> Int
combi n k
  | n < 0 || k < 0 || k > n = 0
  | k == n = 1
  | otherwise = div (div (fact n) (fact (n - k))) (fact k)

{-w
primes :: Integral a => [a]
primes = map fromIntegral $ [2, 3] ++ primes'
 where
  primes' = [5] ++ f 1 7 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] ]

prime n = primes !! n
primes = sieve [2 ..]
sieve (p : xs) = p : sieve [ x | x <- xs, x `mod` p /= 0 ]
-}
fact :: Int -> Int
fact 0 = 1
fact n | n > 0 = n * fact (n - 1)

knappsackWDP :: Int -> [(Int, Int)] -> VU.Vector Int
knappsackWDP maxW [] = VU.replicate (maxW + 1) 0
knappsackWDP maxW ((w, v) : xs) =
  let tailDP = knappsackWDP maxW xs
   in VU.generate (maxW + 1) $ \i ->
        if w <= i
          then max (tailDP VU.! i) (tailDP VU.! (i - w) + v)
          else tailDP VU.! i

knappsackVDP :: Int -> Int -> Int -> [(Int, Int)] -> VU.Vector Int
knappsackVDP maxW maxV n [] = VU.cons 0 $ VU.replicate (n * maxV) maxW
knappsackVDP maxW maxV n ((w, v) : xs) =
  let tailDP = knappsackVDP maxW maxV n xs
   in VU.generate (n * maxV + 1) $ \i ->
        if v <= i
          then min (tailDP VU.! i) (tailDP VU.! (i - v) + w)
          else tailDP VU.! i

knappsackMinVol ::
  AIO.IOArray (Int, Int) Int ->
  VU.Vector Int ->
  VU.Vector Int ->
  Int ->
  Int ->
  IO Int
knappsackMinVol dp volumeVec valueVec dpSize maxVolume = do
  mapM_
    ( \i ->
        mapM_
          ( \v -> do
              let volume = volumeVec VU.! i
                  value = valueVec VU.! i
              newdp <-
                if (v + value) > dpSize
                  then AIO.readArray dp (i, v)
                  else do
                    dpIVm <- AIO.readArray dp (i, v + value)
                    dpIV <- AIO.readArray dp (i, v)
                    return (min (dpIVm - volume) dpIV)
              AIO.writeArray dp (i + 1, v) newdp
          )
          (reverse [0 .. dpSize])
    )
    [0 .. VU.length valueVec - 1]
  foldM
    ( \acc x -> do
        num <- AIO.readArray dp (VU.length volumeVec, x)
        return $ if num <= maxVolume then x else acc
    )
    0
    [0 .. dpSize]

lcsTable :: BS.ByteString -> BS.ByteString -> V.Vector (VU.Vector Int)
lcsTable xs ys = V.scanl step (VU.replicate (m + 1) 0) xs'
  where
    n = BS.length xs
    m = BS.length ys
    xs' = V.generate n $ BSC8.index xs :: V.Vector Char
    ys' = VU.generate m $ BSC8.index ys :: VU.Vector Char
    step :: VU.Vector Int -> Char -> VU.Vector Int
    step v x = VU.scanl innerStep 0 (VU.zip3 v (VU.tail v) ys')
      where
        innerStep :: Int -> (Int, Int, Char) -> Int
        innerStep a (b, c, y)
          | x == y = 1 + b
          | otherwise = max a c

{-
main = do
xs <- BS.getLine
ys <- BS.getLine
print $ VU.last $ V.last $ lcsTable xs ys
-}

myFilter :: (a -> Bool) -> [a] -> ([a], [a])
myFilter _pred [] = ([], [])
myFilter pred (x : xs)
  | pred x = let (l, r) = myFilter pred xs in (x : l, r)
  | otherwise = let (l, r) = myFilter pred xs in (l, x : r)

--UnionFindデータ構造

type UF_PARENT = VUM.IOVector Int

type UF_RANK = VUM.IOVector Int

type UF = (UF_PARENT, UF_RANK)

uf_makeSet :: Int -> IO UF
uf_makeSet n = do
  parent <- VU.thaw $ VU.generate n id
  rank <- VUM.replicate n 0
  return (parent, rank)

uf_union :: UF -> Int -> Int -> IO ()
uf_union unionFind@(parents, ranks) x y = do
  xRoot <- uf_find unionFind x
  yRoot <- uf_find unionFind y

  xRootRank <- VUM.read ranks xRoot
  yRootRank <- VUM.read ranks yRoot
  if
      | xRootRank > yRootRank -> VUM.write parents yRoot xRoot
      | xRootRank < yRootRank -> VUM.write parents xRoot yRoot
      | xRoot /= yRoot -> do
        VUM.write parents yRoot xRoot
        VUM.write ranks xRoot (xRootRank + 1)
      | otherwise -> return ()

uf_find :: UF -> Int -> IO Int
uf_find unionFind@(parents, ranks) x = do
  xParent <- VUM.read parents x
  if xParent == x
    then return x
    else do
      xRootParent <- uf_find unionFind xParent
      VUM.write parents x xRootParent
      return xRootParent

--UnDirectedGraph(無向グラフ)データ構造

type UDG = M.Map Int [Int]

list2UDG :: [(Int, Int)] -> M.Map Int [Int]
list2UDG =
  foldl
    ( \acc x ->
        let lr = M.insertWith (++) (fst x) [snd x] acc
         in M.insertWith (++) (snd x) [fst x] lr
    )
    M.empty