# Project Euler Problem 23 Solution

## Question

A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.

A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.

As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

import Data.List (sort, group, union)
import Data.Array

pairwise :: (a -> a -> a) -> [a] -> [a]
pairwise f (xs:ys:t) = f xs ys : pairwise f t
pairwise _ t = t

primes :: [Int]
primes = 2 : _Y ((3 :) . gaps 5 . _U . map (\p-> [p*p, p*p+2*p..]))
where
_Y g = g (_Y g)                      -- recursion, Y combinator
_U ((x:xs):t) = x : (union xs . _U . pairwise union) t   -- ~= nub.sort.concat
gaps k s@(x:xs)
| k < x     = k : gaps (k+2) s    -- ~= [k,k+2..]\\s, when
| otherwise =     gaps (k+2) xs   --  k <= head s && null(s\\[k,k+2..])

factorize :: Int -> [Int]
factorize n = primeFactors n primes where
primeFactors 1 _ = []
primeFactors _ [] = []
primeFactors m (p:ps) | m < p * p = [m]
| r == 0 = p : primeFactors q (p:ps)
| otherwise = primeFactors m ps
where (q, r) = quotRem m p

primePowers :: Int -> [(Int, Int)]
primePowers n = [(head x, length x) | x <- group $factorize n] divisors :: Int -> [Int] divisors n = filter (<n)$ map product $sequence [take (k+1)$ iterate (p*) 1 | (p, k) <- primePowers n]

upperBound :: Int
upperBound = 20161

abundant :: Int -> Bool
abundant n = (sum . divisors) n > n

abundantsArray :: Array Int Bool
abundantsArray = listArray (1, upperBound) $map abundant [1..upperBound] abundants :: [Int] abundants = filter (abundantsArray !) [1..upperBound] remainders :: Int -> [Int] remainders x = map (x-)$ takeWhile (<= x quot 2) abundants

sums :: [Int]
sums = filter (any (abundantsArray !) . remainders) [1..upperBound]

main :: IO ()
main = print $sum [1..upperBound] - sum sums $ ghc -O2 -o abundant abundant.hs
$time ./abundant real 0m0.065s user 0m0.064s sys 0m0.000s ## Python #!/usr/bin/env python3 import math from collections import defaultdict from itertools import * from functools import reduce def factorize(n): if n < 1: raise ValueError('fact() argument should be >= 1') if n == 1: return [] # special case res = [] # iterate over all even numbers first. while n % 2 == 0: res.append(2) n //= 2 # try odd numbers up to sqrt(n) limit = math.sqrt(n+1) i = 3 while i <= limit: if n % i == 0: res.append(i) n //= i limit = math.sqrt(n+i) else: i += 2 if n != 1: res.append(n) factors = sorted(res) histogram = defaultdict(int) for factor in factors: histogram[factor] += 1 return list(histogram.items()) def divisors(n): factors = factorize(n) nfactors = len(factors) f = [0] * nfactors while True: yield reduce(lambda x, y: x*y, [factors[x][0]**f[x] for x in range(nfactors)], 1) i = 0 while True: f[i] += 1 if f[i] <= factors[i][1]: break f[i] = 0 i += 1 if i >= nfactors: return def proper_divisors(n): return list(divisors(n))[:-1] def classify(n): total = sum(proper_divisors(n)) if total == n: # perfect return 0 elif total > n: # abundant return 1 else: # deficient return -1 def main(): abundant = set(number for number in range(2, 30000) if classify(number) == 1) sums = sorted(set(sum(c) for c in combinations_with_replacement(abundant, 2))) print((sum(number for number in range(1,30000) if number not in sums))) if __name__ == "__main__": main() $ time python3 abundant.py
real   0m15.689s
user   0m15.560s
sys    0m0.016s

## Ruby

#!/usr/bin/env ruby
require 'mathn'

class Integer
def divisors
return [1] if self == 1
primes, powers = self.prime_division.transpose
exponents = powers.map{|i| (0..i).to_a}
divisors = exponents.shift.product(*exponents).map do |powers|
primes.zip(powers).map{|prime, power| prime ** power}.inject(:*)
end
divisors.take divisors.length - 1
end

def abundant?
self.divisors.reduce(:+) > self
end
end

abundants = (1..28213).select { |n| n.abundant? }
i = 0
sums = []
abundants.each do |x|
abundants[i..abundants.length].each do |y|
sum = x + y
sums << sum unless sum > 28213
end
i += 1
end
sums.uniq!
puts (1..28213).reject { |n| sums.include? n }.reduce(:+)
\$ time ruby abundant.rb
real   0m4.137s
user   0m4.060s
sys    0m0.028s