Problem:
Score : 800 points

Problem Statement

You are given a polynomial of degree N with integer coefficients: f(x)=a_Nx^N+a_{N-1}x^{N-1}+...+a_0. Find all prime numbers p that divide f(x) for every integer x.

Constraints

0 \leq N \leq 10^4

|a_i| \leq 10^9(0\leq i\leq N)

a_N \neq 0

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_N
:
a_0

Output

Print all prime numbers p that divide f(x) for every integer x, in ascending order.

Sample Input 1

2
7
-7
14

Sample Output 1

2
7

2 and 7 divide, for example, f(1)=14 and f(2)=28.

Sample Input 2

3
1
4
1
5

Sample Output 2

There may be no integers that satisfy the condition.

Sample Input 3

0
998244353

Sample Output 3

998244353

Solution in Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

{-# LANGUAGE
  ScopedTypeVariables, BangPatterns, TupleSections, ExplicitForAll,
  LambdaCase, MultiWayIf, Unsafe, RecordWildCards, FlexibleContexts, CPP,
  NoMonomorphismRestriction, GADTs, PatternGuards,
  RankNTypes, EmptyDataDecls, EmptyCase, ViewPatterns #-}


import Data.Bits
import Data.List
import Data.Maybe
import Data.Tuple
import Data.Ord
import Data.Int
import Data.Word
import Data.Char
import Data.Function
import Data.STRef
import Data.IORef
import Data.Monoid
import Data.Functor
import System.IO
import Control.Arrow
import Control.Applicative
import Control.Monad
import Control.Monad.Primitive
import Control.Monad.State.Strict
import Control.Monad.ST
import Control.Monad.ST.Lazy (strictToLazyST, lazyToStrictST)
import qualified Control.Monad.ST.Lazy as STL
-- import Control.Monad.ST.Safe
import Control.DeepSeq
import qualified Data.ByteString.Char8 as BS
import qualified Data.ByteString.Lazy.Char8 as BSL
import qualified Data.ByteString.Builder as BSB
import Data.IntMap (IntMap)
import qualified Data.IntMap.Strict as IMS
import qualified Data.IntMap.Lazy as IML
import Data.IntSet (IntSet)
import qualified Data.IntSet as IS
import qualified Data.Sequence as Seq
import Data.Sequence (Seq)
import qualified Data.Array.IArray as A
import qualified Data.Array.MArray.Safe as A
import qualified Data.Array.MArray as A
import Data.Array (Array)
import Data.Array.Unboxed (UArray)
import Data.Array.IArray (IArray)
import Data.Array.MArray.Safe (MArray)
import Data.Array.IO.Safe (IOArray, IOUArray)
import Data.Array.ST.Safe (STArray, STUArray, runSTArray, runSTUArray)
import qualified Data.Vector as V
import qualified Data.Vector.Mutable as VM
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import qualified Data.Vector.Primitive as VP
import qualified Data.Vector.Primitive.Mutable as VPM
import qualified Data.Vector.Generic as VG
import qualified Data.Vector.Generic.Mutable as VGM
import GHC.Exts (build)

main :: IO ()
main = do
  n <- readInt <$> getLine
  asRev <- VU.unfoldrN (n+1) (runStateT rInt) <$> BSL.getContents
  let f0 = VU.last asRev
      f1 = VU.sum asRev
      fm1 = VU.foldl' (\ !val !coe -> coe - val) 0 asRev
      prms = map fst $ primeDecp $ f0 `gcd` f1 `gcd` fm1
  putStr $ unlines $ map show
    $ [ p | p <- prms,
            all (\x -> evalMod asRev x p == 0) [2..n*2]]

evalMod :: VU.Vector Int -> Int -> Int -> Int
evalMod polyRev !x !modulus
  = VU.foldl' (\ !val !coe -> (val * x + coe) `mod` modulus) 0 polyRev

{-# INLINE primeDecp #-}
primeDecp :: Int -> [(Int,Int)]
primeDecp n = build $ \cons nil ->
  let (!expt2,!n1) = countExpt n 2
      loop !p !n !rt | n == 1    = nil
                     | p > rt    = cons (n,1) nil
                     | expt > 0  = cons (p,expt) $ loop (p+2) n_ (sqrtInt n_)
                     | otherwise = loop (p+2) n rt
        where
          (expt,n_) = countExpt n p
  in (if expt2 > 0 then cons (2,expt2) else id)
     $ loop 3 n1 (sqrtInt n1)
    


{-# INLINE countExpt #-}
countExpt :: Int -> Int -> (Int, Int)
countExpt !x !base
  = (\(Just (!j,!q0,!q1,!r1)) -> (j,q0))
      $ find (\(!j,!q0,!q1,!r1) -> r1 /= 0) divlist
  where
    (!q0,!r0) = x `divMod` base
    divlist = iterate (\(!j,!q0,!q1,!r1) -> let (q2,r2) = q1 `divMod` base
                                            in (j+1,q1,q2,r2))
              (0,x,q0,r0)

{-# INLINE sqrtInt #-}
sqrtInt :: Int -> Int
sqrtInt x = ceiling $ sqrt (fromIntegral x :: Double) 


rInt :: StateT BSL.ByteString Maybe Int
rInt = StateT $ BSL.readInt . BSL.dropWhile (<'!')
rIntS :: StateT BS.ByteString Maybe Int
rIntS = StateT $ BS.readInt . BS.dropWhile (<'!')

#define D(f,r,d) f::Integral a=>a->d;f=fromIntegral;r::String->d;r=read
#define C(f,r,g,h,d) D(f,r,d);g,h::RealFrac a=>a->d;g=floor;h=ceiling
C(_toInteger_,readInteger,floorInteger,ceilInteger,Integer)
C(toInt,readInt,floorInt,ceilInt,Int)
C(toI8,readI8,floorI8,ceilI8,Int8)
C(toI16,readI16,floorI16,ceilI16,Int16)
C(toI32,readI32,floorI32,ceilI32,Int32)
C(toI64,readI64,floorI64,ceilI64,Int64)
C(toWord,readWord,floorWord,ceilWord,Word)
C(toW8,readW8,floorW8,ceilW8,Word8)
C(toW16,readW16,floorW16,ceilW16,Word16)
C(toW32,readW32,floorW32,ceilW32,Word32)
C(toW64,readW64,floorW64,ceilW64,Word64)
D(toDouble,readDouble,Double)
D(toFloat,readFloat,Float)
#undef D
#undef C

{-# INLINE wrA #-}
wrA = A.writeArray
{-# INLINE rdA #-}
rdA = A.readArray
{-# INLINE mdA #-}
mdA arr f i = do
  ai <- rdA arr i
  let fai = f ai 
  wrA arr i fai
  return (ai,fai)
{-# INLINE mdA' #-}
mdA' arr f i = do
  !ai <- rdA arr i
  let !fai = f ai
  wrA arr i fai
  return (ai,fai)
{-# INLINE swapA #-}
swapA arr i j = do
  ai <- rdA arr i
  wrA arr i =<< rdA arr j
  wrA arr j ai

#define N(f,g,a,m)\
  f :: forall e i s. (C(a,m)A.Ix i) => (i,i) -> e -> m (a i e); f=A.newArray;\
  g :: forall e i s. (C(a,m)A.Ix i) => (i,i) -> m (a i e); g=A.newArray_
#define C(a,m)
N(newIOA,newIOA_,IOArray,IO)
N(newSTA,newSTA_,STArray s,ST s)
#undef C
#define C(a,m) MArray (a) e (m), 
N(newIOUA,newIOUA_,IOUArray,IO)
N(newSTUA,newSTUA_,STUArray s,ST s)
#undef C
#undef N