Just some tips to make for faster and more concise code:
In your bool abundant(int):
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if(factors_sum > num)
return true;
return false;

Also do not check all the factors in the number. Just check all the numbers within a range equal to the square root of the number you are checking. i.e.
line 56:
for(int i = 1; i < sqrt(num) ;)
Or
for(int i = 1; i < pow(num, 0.5) ;)
I don't understand what you are doing in line 20. Why are you setting all the previous values to 2 wouldn't this just in the end fill every spot in the array with 2 except for the last value stored?
A few more words:
1. The question is asking you to find the sum of all the numbers
below 28123 which cannot be written as the sum of 2 abundant numbers.
2. So if I were solving this question, I would first get all the abundant numbers who their sum is less than 28123. i.e. fill an array with 24, 32, 36, 40, 48, etc.
3. Then I would create a nested forloop. The first forloop in the nest will go through the values 128123 and the forloop below will go through the array of abundant sums.
if the value in the first forloop is not equal to the value in the one below, continue adding the numbers from the first forloop
else skip that number and continue
Also try using dynamically allocated arrays:
int *Array = new int[1];
and you can resize it as you wish, you can use this function if you want:
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int * Resize(int arr[], int length, int val)
{
int * array2 = new int [length + 1];
for (int J = 0; J < length; ++J)
array2[J] = arr[J];
array2[length] = val;
delete[] arr;
arr = NULL;
return array2;
}

ex
Array = Resize(Array, size, num)