Project Euler Problem 23

Problem 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.

# 找到所有不能写作两个富余数之和的正数，求它们的和。富余数是指因子之和大于自身的数。通过数学分析已知大于28123的正数都可以写作两个富余数的和。

def isabundant(n):
    sum1 = 0
    for i in range(1,int(n**0.5)+1):
        if n % i == 0:
            sum1 += (i + n/i)
        if n / i == i:
            sum1 -= i
        if sum1 > 2*n:
            return True
    else:
        return False

sum1 = sum(range(28123+1))
sum2 = 0
set1 = set()
for i in range(1,28123+1):
    if isabundant(i):
        set1.add(i)
    for j in set1:
        if (i-j) in set1:
            sum2 += i
            break
print(sum2)
print(sum1 - sum2)

结果：4179871