'''
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.
'''

def divisors(start, num):

    sum = 0

    for v in xrange(start, 0, -1):

        if num % v == 0:
            sum += v

    return(sum)

def abundant(usl):

    soln = []

    for num in xrange(usl + 1):

        sum_divs = divisors(num - 1, num)

        if sum_divs > num:
            soln.append(num)

    return(soln)

def abundant_sums(arr):
    soln = set()

    for i in xrange(len(arr)):

        for j in xrange(i, len(arr)):
            soln.add(arr[i] + arr[j])


    return(soln)

if __name__ == '__main__':
    usl = 28123

    abndnt_nums = abundant(usl)

    abndnt_sums = abundant_sums(abndnt_nums)

    ints = set(range(usl))

    print('Sum of all positive integers that' \
            'are not the sum of two abundant numbers is' \
            .format(sum(ints - abndnt_sums)))
