
from eutility.eusequence import abundant
from itertools import takewhile
from itertools import product

def euler023(limit):
    ''' Non-abundant sums

    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. '''    
    return sum(set(xrange(limit)) - set(sum(x) for x in product(takewhile(lambda x: x < limit, abundant()), repeat=2)))
