33 lines
1.3 KiB
Haskell
33 lines
1.3 KiB
Haskell
-- A perfect number is a number for which the sum of its proper divisors is
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-- exactly equal to the number. For example, the sum of the proper divisors of
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-- 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect
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-- number.
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--
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-- A number n is called deficient if the sum of its proper divisors is less
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-- than n and it is called abundant if this sum exceeds n.
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--
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-- As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest
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-- number that can be written as the sum of two abundant numbers is 24. By
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-- mathematical analysis, it can be shown that all integers greater than 28123
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-- can be written as the sum of two abundant numbers. However, this upper limit
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-- cannot be reduced any further by analysis even though it is known that the
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-- greatest number that cannot be expressed as the sum of two abundant numbers
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-- is less than this limit.
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--
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-- Find the sum of all the positive integers which cannot be written as the sum
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-- of two abundant numbers.
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import Data.Maybe
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import Euler
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isAbundant :: Int -> Bool
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isAbundant n = sum (properDivisors n) > n
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abundant :: [Int]
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abundant = filter isAbundant [1..]
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abundantPair :: Int -> Maybe (Int, Int)
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abundantPair n = listToMaybe [ (p,q) | p <- takeWhile (<= (n `div` 2)) abundant, let q = n - p, isAbundant q ]
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main = print $ sum $ filter (isNothing . abundantPair) [1..28123]
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