Almost-almost Perfect Numbers

Question

What are the integers $n>1$, if any, whose proper factors sum to $n+1$?

In other words, whose factors sum to $2n+1$.

I tested for $n<10,000,000$ in Mathematica and found no such number. Yet the sum of proper factors fluctuates above and below $n+1$, so I can't see any obvious reason why such a number shouldn't exist.

Answer 1

A note at the OEIS (https://oeis.org/A033880) indicates:

For no known n is a(n) = 1. If there is such an n it must be greater than 10^35 and have seven or more distinct prime factors (Hagis and Cohen 1982). - Jonathan Vos Post, May 01 2011

The citation is to P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982, which you can view online here. They refer to numbers whose factors sum to $2n+1$ as "quasiperfect" numbers and prove the facts given above.

(I've assumed that you mean the proper factors sum to $n+1$. If $n$ is a power of two, then its proper factors sum to $n-1$, so presumably a computer search should have found a bunch of answers.)