Project Euler Problem 023

# 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)
```