Motivation: In group theory it is proved that the center of a p-group is non-trivial. Essentially the proof is making use of the fact that for any number of the form $n=p^k$ where $p$ is prime, there is no way to represent $n-1$ as a sum of prime factors of $n$ (because all factors divide $p$ while $n-1$ does not)
Question: Are there any other numbers $n$ such that $n-1$ cannot be written as a sum of non-trivial factors of $n$, allowing for repetitions?
I suspect that the answer is no because there seems to be too many ways of playing with the factors in the sum once there are two of them. I was also able to prove there is no such $n$ in a few simple cases but I think there might be a proof for the general case as well.
Claim:
If $n$ is a positive integer with at least two distinct prime factors, then $n-1$ can be expressed as a sum of nontrivial (i.e., factors not equal to $1$) positive integer factors of $n$ (possibly repeated).
Proof:
Since $n$ has at least two distinct prime factors, we can write $n=ab$, where $a,b$ are integers with $a,b > 1$, and $\gcd(a,b)=1$.
It follows by a well known elementary result (and easily proved), that every integer $m$ greater than $ab-a-b$ can be expressed as $m=ax+by$, for some nonnegative integers $x,y$.
Now simply note that $n=ab$ implies $n-1 > ab-a-b$.