将 0 添加到 Project Euler 问题 23 “Non-Abundant Sums” 的列表中会改变答案

Question

I'm a beginner in programming, and I'm trying to practice Project Euler questions. Question 23:

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.

I got the correct answer by the following code:

list1=[]
list2=[]
for num in range(1,28123):
  list1.append(num)
for num in list1:
  sum1=1
  for i in range (2,int(num**0.5)+1):
    if not num%i:
      sum1+=(i+(num/i))
  if int(num**0.5)==(num**0.5):
    sum1-=(num**0.5)
  if sum1>num:
    list2.append(num)
for i in range(len(list2)):
  for j in range(i,len(list2)):
    check=list2[i]+list2[j]
    if check<28123:
      list1[check-1]=0
print(sum(list1))

I tried to make a small modification to this code, by changing the definition of list1 by adding a '0' by the following code:

list1=[x for x in range (28123)]
list2=[]
for num in list1:
  sum1=1
  for factor in range (2,int(num**0.5)+1):
    if not num%factor:
      sum1+=(factor+(num/factor))
  if int(num**0.5)==(num**0.5):
    sum1-=(num**0.5)
  if sum1>num:
    list2.append(num)
for i in range(len(list2)):
  for j in range(i,len(list2)):
    check=list2[i]+list2[j]
    if check<28123:
      list1[check]=0
print(sum(list1))

As you can see, I also changed the second last line according the new indexes. But this changed my answer from 4179871 to 4178876. I can't understand why. The former is the correct answer. I know my code is not very efficient as I'm a beginner, trying to improve. Help will be much appreciated!

Edit:

I tried to check if 0 is getting added into the abundant number's list2, but the following code gave me [] as an output, hence it wasn't.

num = 0
list2 = []
sum1 = 1
for i in range (2, int (num ** 0.5) + 1):
    if not num % i:
        sum1 += (i+ (num / i))
    if int (num ** 0.5) == (num ** 0.5):
        sum1 -= (num ** 0.5)
    if sum1 > num:
        list2.append(num)
print(list2)

Answer 1

You inserted 0; your code recognizes this as an abundant number. Therefore, any abundant number that was previously not the sum of two others... is suddenly the sum of itself and 0. It's excluded from the list.

I strongly recommend that you learn to perform basic logical debugging. I determined this with a simple insertion. I pasted both of your algorithms into a file. In between, I saved the results of the first algorithm:

...
print(sum(list1))
keep1 = list1
keep2 = list2

Then, after the second algorithm, I simply found the differences between the lists you formed in the two algorithms:

print("acquired abundant numbers", set(list2) - set(keep2))
print("lost summed numbers", set(keep1) - set(list1))

Output:

4179871
4178876
acquired abundant numbers {0}
lost summed numbers {945, 18, 12, 20}

... and there is the telling evidence: three numbers less than 24, which is stated as the lowest number qualifying for the summation. Each of these is abundant, and the first added output line shows that your code mis-allocated 0 as abundant.

This is one of the most basic debugging techniques: when you have a solid difference in expected and actual output, print or otherwise compare the intermediate values to find the point of difference.

You note that I'm not tracking this into your code: you didn't make it easy to read. See the PEP-8 guidelines for suggestions.