Let $n$ be a positive integer with $n\equiv 4\mod 6$ and define $p:=\frac{n^2+n+1}{3}$ (which is in this case a positive integer as well).
Conjecture : $$p^2\mid n^n+(n+1)^{n+1}$$ for every $n$ of the given form.
The conjecture is true upto $n=10^9$
Trial : We have to show $(n^2+n+1)^2 \mid 9(n^n+(n+1)^{n+1})$ and with $x^6\equiv (x+1)^6\equiv 1\mod (x^2+x+1)$ I could show $x^2+x+1\mid x^n+(x+1)^{n+1}$ , but I did not manage to find the general remainder of $9(x^n+(x+1)^{n+1})$ modulo $(x^2+x+1)^2$ to finish the proof. Can "lifting the exponent" help here ?