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..]

-- 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 (\(_,y) -> isPrime y) $ map (\x -> (x,x+2)) primes
				
-------------------------------------------------------------------------------

-- 2

-- generates powers of 2	
pof2s :: [Integer]
pof2s = [1] ++ map (2*) pof2s
	
pow :: Int -> Integer
pow 0 = 1
pow n = pow (n-1) + pow (n-1)

-- 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)

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

-- stream of faculties
facGen :: [Integer]
facGen = map (fac) [1..]

-- 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) [1..(fromIntegral k)]
	
-- helper function h using mem-table for the power-series (z^i) and for faculty (i!)
hMT :: Integer -> Integer -> Float
hMT z i = (fromInteger $ pofNs z !! (fromInteger i)) / (fromInteger $ facGen !! ((fromIntegral i)-1))

-- 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..]