Problem 23

Non-abundant sums

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.

非盈数和

完全数是指真约数之和等于自身的数。例如，$28$的真约数之和为$1 + 2 + 4 + 7 + 14 = 28$，因此$28$是一个完全数。

若一个数$n$的真约数之和小于$n$，则称之为亏数；反之，则称之为盈数。

由于$12$是最小的盈数（它的真约数之和为$1 + 2 + 3 + 4 + 6 = 16$），所以能够表示成两个盈数之和的最小数是$24$。通过数学分析可以得出，所有大于$28123$的数都可以被表示成两个盈数的和。但是，这仅仅是通过数学分析所能得到的最好上界，而实际上不能被表示成两个盈数之和的最大数要小于这个值。

求所有不能被表示成两个盈数之和的正整数之和。