79 lines
2.1 KiB
Haskell
79 lines
2.1 KiB
Haskell
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import Data.List (nub)
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binomial :: (Integral n) => n -> n -> n
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binomial n 0 = 1
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binomial n k
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| n == k = 1
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| otherwise = binomial (n - 1) k + binomial (n - 1) (k - 1)
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-----------------------------------------------
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binomial2 :: Int -> Int -> Int
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binomial2 n k = triangle !! n !! k
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where
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triangle = iterate next_row [1]
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next_row row = zipWith (+) ([0] ++ row) (row ++ [0])
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-----------------------------------------------
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merge :: Ord a => [a] -> [a] -> [a]
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merge xl [] = xl
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merge [] yl = yl
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merge (x:xl) (y:yl)
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| x <= y = x:merge xl (y:yl)
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| otherwise = y:merge (x:xl) yl
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mergesort :: Ord a => [a] -> [a]
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mergesort [] = []
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mergesort [a] = [a]
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mergesort xl = merge (mergesort (first_half xl)) (mergesort (second_half xl))
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first_half xl = let { n = length xl } in take (div n 2) xl
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second_half xl = let { n = length xl } in drop (div n 2) xl
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------------------------------------------------
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gcde :: (Integral a) => a -> a -> (a, a, a)
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gcde 0 b = (b, 0, 1)
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gcde a b =
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let (g, x, y) = gcde (mod b a) a in (g, y - (div b a) * x, x)
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de :: (Integral a) => a -> a -> (a, a, a)
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de a b = let (g, x, y) = gcde a b in (x, y, g)
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------------------------------------------------
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prime_factors :: (Integral a) => a -> [a]
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prime_factors n = factorize n 2
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factorize :: (Integral a) => a -> a -> [a]
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factorize 1 _ = []
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factorize n factor
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| mod n factor == 0 = factor : factorize (div n factor) factor
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| otherwise = factorize n (factor + 1)
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------------------------------------------------
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_gcd :: Integral a => a -> a -> a
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_gcd 0 b = b
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_gcd a b = _gcd (mod b a) a
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is_coprime :: Integral a => a -> a -> Bool
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is_coprime a b = _gcd a b == 1
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totient :: Integral a => a -> a
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totient n = fromIntegral (length (filter (is_coprime n) [1..n]))
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------------------------------------------------
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totient2 :: Integral a => a -> a
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totient2 n = foldl (\acc p -> div (acc * (p - 1)) p) n (nub (prime_factors n))
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------------------------------------------------
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primes :: Integral a => a -> [a]
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primes n = sieve [2..n]
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where
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sieve [] = []
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sieve (x:xs) = x : sieve [y | y <- xs, mod y x /= 0]
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