Let $n$ be an odd number, $x,y$ integers and $p$ a prime number. Now, suppose that $p\ne n$ and $$ p|\frac{x^n+y^n}{x+y} $$
Then, I have been observed that $p \equiv 1 \pmod{n}$. This is, all of the prime factors of the previous expression are of the form $p=nq+1$ An easy way to see this with numerical examples is to choose $n=5$, then the last digit of each prime number will be $1$. Examples of this case are: $$ \frac{5^5+8^5}{5+8}=11\cdot 251 $$ $$ \frac{19^5+26^5}{19+26}=5\cdot 11\cdot 1801 $$ $$ \frac{113^5+257^5}{113+257}=5\cdot 615988781\ $$
I post it as a conjecture, although I ignore if it's a know theorem. I have been searching on the Internet, where I haven't found any interesting about it.
Note: this conjecture seems to be true also with: $$ p|\frac{x^n-y^n}{x-y} $$
The answer to your conjecture and much, much more about the form(s) $$ \frac{a^n \pm b^n}{a \pm b}, \qquad a,b,n \in \mathbb{Z} $$ and its factors can be found in Ribenboim’s wonderful book, Fermat’s Last Theorem for Amateurs, specifically Chapter II.