## 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.

### To solve this problem, we first need to find all the abundant numbers up to 28123. Then we need to find all numbers that can be expressed as the sum of two abundant numbers. Finally, we need to find all numbers that cannot be expressed this way and their sum.

Here is a step by step explanation on how we do this:

Step 1:

Find all the abundant numbers up to 28123.

The most difficult part of this is finding the proper divisors of a number. Note that abundant numbers are fairly common – roughly 25% of numbers up to 28123 are abundant. We use a divisor function which sums all divisors of a number.

Step 2:

Mark all positive integers which can be written as the sum of two abundant numbers.

To do this we iterate through each of the abundant numbers we found in Step 1. For each number, we find all pairs that sum up to less than 28123. For each of these pairs, we mark that they can be represented as the sum of two abundant numbers.

Step 3:

Finally, we iterate through all the numbers up to 28123, and if they can’t be represented as the sum of two abundant numbers, we add them to our sum.

This is a computationally expensive problem. If you try to find a more mathematical solution, you’ll find that you can’t reduce the upper limit any further. But we already know that all integers greater than 28123 can be written as the sum of two abundant numbers. So, we will use computational power to solve this problem.

Let’s solve this problem with python code:

“`python

import numpy as np

MAX = 28123

divisor_sums = np.zeros(MAX+1, dtype=int)

for i in range(1, MAX+1):

divisor_sums[2*i::i] += i

# Create a boolean array “abundant[:n+1]”

abundant = divisor_sums > np.arange(MAX+1)

# All the numbers up to 28123 that are sums of two abundant numbers

is_abundant_sum = np.zeros(MAX+1, dtype=bool)

for i in range(1, MAX+1):

if abundant[i]:

is_abundant_sum[i + np.arange(i, min(MAX – i, MAX//2)+1)] = True

print(np.where(~is_abundant_sum)[0].sum())

“`

In this program, we first compute the divisor sums, then determine if these numbers are abundant or not. After finding the abundant numbers, we mark the sums of abundant numbers. Finally, we calculate the sum of numbers which cannot be represented as the sum of two abundant numbers.

When we run this program, we get the desired result which is the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

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