TITLE: Has $n^{n+1}+(n+1)^{n+2}$ other obvious factors than that I found?
QUESTION [7 upvotes]: Has the number $$f(n):=n^{n+1}+(n+1)^{n+2}$$ "obvious" factors (algebraic, aurifeuillan or similar kinds) apart from those , I mention below ?

I only managed to find out forced factors for odd numbers $\ n\ $ :

If $\ n\ $ is of the form $\ 6k+1\ $ , then $\ f(n)\ $ is divisible by $\ 3\ $. 
If $\ n\ $ is of the form $\ 6k+3\ $ , then $\ f(n)\ $ is divisible by $\ n^2+n+1\ $ 
and finally, if $\ n+2\ $ is prime, then $\ f(n)\ $ is divisble by $\ n+2\ $. 

For even $n$, I did not find forced factors.
The smallest not completely factored number of this form is $f(62)$. It has the composite cofactor $$29645851324749161395794060252012567916992450650017954$$
$$8416412620499302880901240095492001218810908429181608669479$$ with $111$ digits.

REPLY [4 votes]: The full factorization of $f(62)$ is
$$\begin{eqnarray}
f(62) & = & 97\times 503\times 434254837008211200040837849611255155960657\times \\ & & 682683272545530598298287751380982048464896752734487326295497775449847\end{eqnarray}$$
Somewhat disappointingly, it was reached using raw computing power rather than any smart ideas; just by running YAFU, the factoring tool.