Define a Perfectly Abundant Number as a number with the property: $$ x : \sigma(x) = 2x + 1, $$
where $\sigma$ is the sum of the factors of $x$, including itself. In other words, $x$ isn't a perfect number because the sum of its proper factors (not including $x$) is one greater than itself, making it Abundant. However, it is almost perfect because it differs by only one. Are the there any such number. In a brute force search with sage I didn't find any less than $10^6$. Is there a mathematical proof that no such number exists?