# Statement

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.

# Solution

The solution is straightforward.

from CommonFunctions import factors def is_abundant(n): lst = factors(n) lst.remove(n) if sum(lst) > n: return True return False def is_sum_of_abundants(n, all_abundants, set_all_abundants): i = 0 while all_abundants[i] <= n // 2: if (n - all_abundants[i] in set_all_abundants): return True i += 1 return False if __name__ == '__main__': all_abundants = [i for i in range(12, 28124) if is_abundant(i)] set_all_abundants = set(all_abundants) result = sum(i for i in range(1, 28124) if not is_sum_of_abundants(i, all_abundants, set_all_abundants)) print("The result is:", result)

The Python file is available for download here.