module ProjectEuler
( isPrime
, digitSum
, sumProperDivisors
, countDivisors
) where

import Data.Char (digitToInt)
import Data.List (nub)

isPrime :: (Integral n) => n -> Bool
isPrime 1 = False
isPrime 2 = True
isPrime 3 = True
isPrime n =
    n `mod` 2 /= 0 && n `mod` 3 /= 0 && null [ x | x <- candidates, n `mod` x == 0 || n `mod` (x+2) == 0 ]
    where candidates = [5,11..limit]
          limit = floor(sqrt(fromIntegral n)) + 1


digitSum :: (Integral a, Show a) => a -> Int
digitSum n = sum $ map digitToInt $ show n


sumProperDivisors :: (Integral a) => a -> a
sumProperDivisors n = sum [ if x /= y then x + y  else x | x <- [2..floor $ sqrt $ fromIntegral n], let y = n `div` x, n `mod` x == 0 ] + 1


countDivisors :: (Integral a) => a -> Int
countDivisors n = length $ nub $ concat [ [x, n `div` x] | x <- [1..limit], n `mod` x == 0 ]
    where limit = floor $ sqrt $ fromIntegral n