Problem:
Score : 300 points

Problem Statement

There are N integers a_1, a_2, ..., a_N not less than 1.
The values of a_1, a_2, ..., a_N are not known, but it is known that a_1 \times a_2 \times ... \times a_N = P.

Find the maximum possible greatest common divisor of a_1, a_2, ..., a_N.

Constraints

1 \leq N \leq 10^{12}

1 \leq P \leq 10^{12}

Input

Input is given from Standard Input in the following format:

N P

Output

Print the answer.

Sample Input 1

3 24

Sample Output 1

2

The greatest common divisor would be 2 when, for example, a_1=2, a_2=6 and a_3=2.

Sample Input 2

5 1

Sample Output 2

1

As a_i are positive integers, the only possible case is a_1 = a_2 = a_3 = a_4 = a_5 = 1.

Sample Input 3

1 111

Sample Output 3

111

Sample Input 4

4 972439611840

Sample Output 4

206

Solution in Haskell (GHC 7.10.3) (judge verdict: Accepted)

-- CADDI 2018 C
{-# LANGUAGE BangPatterns #-}
import Control.Monad
import Data.Maybe
import Debug.Trace
import Data.List
import qualified Data.Map.Strict as Map
import qualified Data.IntMap.Strict as IntMap
import qualified Data.Set as Set
import qualified Data.IntSet as IntSet
import Data.Functor
import Data.Array

main :: IO ()
main = do
  (n:p:_) <- fmap (map read . words) getLine :: IO [Integer]
  print $ solve n p

-- solve :: Int -> Int -> Int
solve n p = go 0 cm
  where
    cm = toCountMap $ primeFactors p
    go i cm 
      | Map.null filtered = 1
      | otherwise = (Map.foldrWithKey (\k _ acc -> k * acc) 1 filtered) * go (i+n) filtered
      where 
        filtered = Map.filter (n+i <=) cm

toCountMap :: (Ord k, Integral a) => [k] -> Map.Map k a
toCountMap xs = Map.fromListWith (+) $ zip xs (repeat 1)
   
factorization :: Integer -> [Integer]
factorization 1 = []
factorization x = let v = head $ naiveFactors x
                  in  v : factorization (x `div` v)
naiveFactors :: Integer -> [Integer]
naiveFactors n = [x | x <- [2..n], n `mod` x == 0]


primes :: Integral int => [int]
primes = wheelSieve 6

{-# SPECIALISE primes :: [Int]     #-}
{-# SPECIALISE primes :: [Integer] #-}

-- | 
-- This function returns an infinite list of prime numbers by sieving
-- with a wheel that cancels the multiples of the first @n@ primes
-- where @n@ is the argument given to @wheelSieve@. Don't use too
-- large wheels. The number @6@ is a good value to pass to this
-- function. Larger wheels improve the run time at the cost of higher
-- memory requirements.
-- 
wheelSieve :: Integral int
           => Int    -- ^ number of primes canceled by the wheel
           -> [int]  -- ^ infinite list of primes
wheelSieve k = reverse ps ++ map head (sieve p (cycle ns))
 where (p:ps,ns) = wheel k

{-# SPECIALISE wheelSieve :: Int -> [Int]     #-}
{-# SPECIALISE wheelSieve :: Int -> [Integer] #-}

-- |
-- Checks whether a given number is prime.
-- 
-- This function uses trial division to check for divisibility with
-- all primes below the square root of the given number. It is
-- impractical for numbers with a very large smallest prime factor.
-- 
isPrime :: Integral int => int -> Bool
isPrime n | n > 1     = primeFactors n == [n]
          | otherwise = False

{-# SPECIALISE isPrime :: Int     -> Bool #-}
{-# SPECIALISE isPrime :: Integer -> Bool #-}

-- |
-- Yields the sorted list of prime factors of the given positive
-- number.
-- 
-- This function uses trial division and is impractical for numbers
-- with very large prime factors.
-- 
primeFactors :: Integral int => int -> [int]
primeFactors n = factors n (wheelSieve 6)
 where
  factors 1 _                  = []
  factors m (p:ps) | m < p*p   = [m]
                   | r == 0    = p : factors q (p:ps)
                   | otherwise = factors m ps
   where (q,r) = quotRem m p

{-# SPECIALISE primeFactors :: Int     -> [Int]     #-}
{-# SPECIALISE primeFactors :: Integer -> [Integer] #-}

-- Auxiliary Definitions
------------------------------------------------------------------------------

-- Sieves prime candidates by computing composites from the result of
-- a recursive call with identical arguments. We could use sharing
-- instead of a recursive call with identical arguments but that would
-- lead to much higher memory requirements. The results of the
-- different calls are consumed at different speeds and we want to
-- avoid multiple far apart pointers into the result list to avoid
-- retaining everything in between.
--
-- Each list in the result starts with a prime. To obtain composites
-- that need to be cancelled, one can multiply all elements of the
-- list with its head.
-- 
sieve :: (Ord int, Num int) => int -> [int] -> [[int]]
sieve p ns@(m:ms) = spin p ns : sieveComps (p+m) ms (composites p ns)

{-# SPECIALISE sieve :: Int     -> [Int]     -> [[Int]]     #-}
{-# SPECIALISE sieve :: Integer -> [Integer] -> [[Integer]] #-}

-- Composites are stored in increasing order in a priority queue. The
-- queue has an associated feeder which is used to avoid filling it
-- with entries that will only be used again much later. 
-- 
type Composites int = (Queue int,[[int]])

-- The feeder is computed from the result of a call to 'sieve'.
-- 
composites :: (Ord int, Num int) => int -> [int] -> Composites int
composites p ns = (Empty, map comps (spin p ns : sieve p ns))
 where comps xs@(x:_) = map (x*) xs

{-# SPECIALISE composites :: Int     -> [Int]     -> Composites Int     #-}
{-# SPECIALISE composites :: Integer -> [Integer] -> Composites Integer #-}

-- We can split all composites into the next and remaining
-- composites. We use the feeder when appropriate and discard equal
-- entries to not return a composite twice.
-- 
splitComposites :: Ord int => Composites int -> (int,Composites int)
splitComposites (Empty, xs:xss) = splitComposites (Fork xs [], xss)
splitComposites (queue, xss@((x:xs):yss))
  | x < z     = (x, discard x (enqueue xs queue, yss))
  | otherwise = (z, discard z (enqueue zs queue', xss))
 where (z:zs,queue') = dequeue queue

{-# SPECIALISE splitComposites :: Composites Int -> (Int,Composites Int) #-}
{-# SPECIALISE
    splitComposites :: Composites Integer -> (Integer,Composites Integer) #-}

-- Drops all occurrences of the given element.
--
discard :: Ord int => int -> Composites int -> Composites int
discard n ns | n == m    = discard n ms
             | otherwise = ns
 where (m,ms) = splitComposites ns

{-# SPECIALISE discard :: Int -> Composites Int -> Composites Int #-}
{-# SPECIALISE
    discard :: Integer -> Composites Integer -> Composites Integer #-}

-- This is the actual sieve. It discards candidates that are
-- composites and yields lists which start with a prime and contain
-- all factors of the composites that need to be dropped.
--
sieveComps :: (Ord int, Num int) => int -> [int] -> Composites int -> [[int]]
sieveComps cand ns@(m:ms) xs
  | cand == comp = sieveComps (cand+m) ms ys
  | cand <  comp = spin cand ns : sieveComps (cand+m) ms xs
  | otherwise    = sieveComps cand ns ys
 where (comp,ys) = splitComposites xs

{-# SPECIALISE sieveComps :: Int -> [Int] -> Composites Int -> [[Int]] #-}
{-# SPECIALISE
    sieveComps :: Integer -> [Integer] -> Composites Integer -> [[Integer]] #-}

-- This function computes factors of composites of primes by spinning
-- a wheel.
-- 
spin :: Num int => int -> [int] -> [int]
spin x (y:ys) = x : spin (x+y) ys

{-# SPECIALISE spin :: Int     -> [Int]     -> [Int]     #-}
{-# SPECIALISE spin :: Integer -> [Integer] -> [Integer] #-}

-- A wheel consists of a list of primes whose multiples are canceled
-- and the actual wheel that is rolled for canceling.
--
type Wheel int = ([int],[int])

-- Computes a wheel that cancels the multiples of the given number
-- (plus 1) of primes.
--
-- For example:
--
-- wheel 0 = ([2],[1])
-- wheel 1 = ([3,2],[2])
-- wheel 2 = ([5,3,2],[2,4])
-- wheel 3 = ([7,5,3,2],[4,2,4,2,4,6,2,6])
--
wheel :: Integral int => Int -> Wheel int
wheel n = iterate next ([2],[1]) !! n

{-# SPECIALISE wheel :: Int -> Wheel Int     #-}
{-# SPECIALISE wheel :: Int -> Wheel Integer #-}

next :: Integral int => Wheel int -> Wheel int
next (ps@(p:_),xs) = (py:ps,cancel (product ps) p py ys)
 where (y:ys) = cycle xs
       py = p + y

{-# SPECIALISE next :: Wheel Int     -> Wheel Int     #-}
{-# SPECIALISE next :: Wheel Integer -> Wheel Integer #-}

cancel :: Integral int => int -> int -> int -> [int] -> [int]
cancel 0 _ _ _ = []
cancel m p n (x:ys@(y:zs))
  | nx `mod` p > 0 = x : cancel (m-x) p nx ys
  | otherwise      = cancel m p n (x+y:zs)
 where nx = n + x

{-# SPECIALISE cancel :: Int -> Int -> Int -> [Int] -> [Int] #-}
{-# SPECIALISE
    cancel :: Integer -> Integer -> Integer -> [Integer] -> [Integer] #-}

-- We use a special version of priority queues implemented as /pairing/
-- /heaps/ (see /Purely Functional Data Structures/ by Chris Okasaki).
--
-- The queue stores non-empty lists of composites; the first element
-- is used as priority.
--
data Queue int = Empty | Fork [int] [Queue int]

enqueue :: Ord int => [int] -> Queue int -> Queue int
enqueue ns = merge (Fork ns [])

{-# SPECIALISE enqueue :: [Int]     -> Queue Int     -> Queue Int     #-}
{-# SPECIALISE enqueue :: [Integer] -> Queue Integer -> Queue Integer #-}

merge :: Ord int => Queue int -> Queue int -> Queue int
merge Empty y                        = y
merge x     Empty                    = x
merge x     y     | prio x <= prio y = join x y
                  | otherwise        = join y x
 where prio (Fork (n:_) _) = n
       join (Fork ns qs) q = Fork ns (q:qs)

{-# SPECIALISE merge :: Queue Int     -> Queue Int     -> Queue Int     #-}
{-# SPECIALISE merge :: Queue Integer -> Queue Integer -> Queue Integer #-}

dequeue :: Ord int => Queue int -> ([int], Queue int)
dequeue (Fork ns qs) = (ns,mergeAll qs)

{-# SPECIALISE dequeue :: Queue Int     -> ([Int],     Queue Int)     #-}
{-# SPECIALISE dequeue :: Queue Integer -> ([Integer], Queue Integer) #-}

mergeAll :: Ord int => [Queue int] -> Queue int
mergeAll []       = Empty
mergeAll [x]      = x
mergeAll (x:y:qs) = merge (merge x y) (mergeAll qs)

{-# SPECIALISE mergeAll :: [Queue Int]     -> Queue Int     #-}
{-# SPECIALISE mergeAll :: [Queue Integer] -> Queue Integer #-}