TITLE: Almost-almost Perfect Numbers
QUESTION [7 upvotes]: 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.

REPLY [6 votes]: 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.)