# Sum of non-trivial divisors of number equals number itself

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?

• do you mean the perfect numbers? Feb 14, 2018 at 13:33
• I was going to say that, but I note that OP excludes 1 from his definition.
– Matt
Feb 14, 2018 at 13:34
• have you tried a computer search?
– lulu
Feb 14, 2018 at 13:37
• @lulu I'm not a programmer. Feb 14, 2018 at 13:38
• even so. Shouldn't be hard to check up to $100$, at least. Easy to see that $pq$, product of distinct primes can't work, for example. Maybe the product of three primes.
– lulu
Feb 14, 2018 at 13:39

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