module AufgabeFFP2
where

-- 1

-- primality check
isPrime :: Integer -> Bool
isPrime n = n > 1 &&
  foldr (\p r ->
    p*p > n
    || (
      (n `mod` p) /= 0
      && r
    )
  )
  True primes

-- series of primes
primes :: [Integer]
primes = 2:filter isPrime [3,5..]

-- pairs of (p,p+2) | p,p+2 <- primes
-- generate all pairs with map and then filter only the valid ones
-- pair is valid if the second component n is a prime
pps :: [(Integer, Integer)]
pps = filter (\(_,x) -> isPrime x) $ map (\p -> (p,p+2)) primes

-------------------------------------------------------------------------------

-- 2

-- generates powers of 2
pof2s :: [Integer]
pof2s = [1] ++ map (2*) pof2s

-- calculates 2^n
pow :: Int -> Integer
pow 0 = 1
pow n = pow (n-1) + pow (n-1)

-- same as pow but uses memtable for power-calculation
powFast :: Int -> Integer
powFast 0 = 1
powFast n = pof2s !! (n-1) + pof2s !! (n-1)

-------------------------------------------------------------------------------

-- 3

-- power series of N
pofNs :: Integer -> [Integer]
pofNs n = [1] ++ (map (n*) $ pofNs n)

-- stream of factorials
facs :: [Integer]
facs = scanl (*) 1 [1..]

-- factorial function
fac :: Integer -> Integer
fac n = product [1..n]

-- function g with memoization (using hFast)
fMT :: Int -> Int -> Float
fMT z k = g z k hMT

-- function g without memoization (uning hSlow)
f :: Int -> Int -> Float
f z k = g z k h

-- actual function g (converts Int to Integer for more precision)
g :: Int -> Int -> (Integer -> Integer -> Float) -> Float
g z k h = sum $ map (h $ fromIntegral z) [0..(fromIntegral k)]

-- helper function h using mem-table for the power-series (z^i) and for factorial (i!)
hMT :: Integer -> Integer -> Float
hMT z i = (fromInteger $ pofNs z !! (fromInteger i)) / (fromInteger $ facs !! (fromInteger i))

-- helper function h without memoization
h :: Integer -> Integer -> Float
h z i =  (fromInteger $ z^i) / (fromInteger $ fac i)

-------------------------------------------------------------------------------

-- 4

-- gets the digits of an integer as a list
digits :: Integer -> [Integer]
digits x
  | x<=0 = []
  | otherwise = (digits $ x `div` 10)++[x `mod` 10]

-- calculates the goedel-number for the given integer
-- returns 0 for non-positive numbers
gz :: Integer -> Integer
gz n
  | n<=0 = 0
  | otherwise = product $ zipWith (^) primes (digits n)

-- goedel-number generator
gzs :: [Integer]
gzs = map gz [1..]