Sum of non-trivial divisors of number equals number itself

Question

Is there a number $n$ such that it equals the sum of its non-trivial divisors (i.e. all of its divisors except 1 and $n$)? If yes, what are such numbers called and what are some examples of them?

I have not found any answers on the internet.

Answer 1

Congratulations, you've walked into an open problem. Any number $n$ would satisfy $\sigma(n)-n-1=n$ or $\sigma(n)=2n+1$, i.e. have an abundance of 1. But a note on the relevant OEIS entry, A033880 (abundance of $n$), states:

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

These are called quasiperfect numbers.