Has $n^{n+1}+(n+1)^{n+2}$ other obvious factors than that I found? 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.
The full factorization of $f(62)$ is

$$\begin{eqnarray}
f(62) &amp; = &amp; 97\times 503\times 434254837008211200040837849611255155960657\times \\ &amp; &amp; 682683272545530598298287751380982048464896752734487326295497775449847\end{eqnarray}$$

Somewhat disappointingly, it was reached using raw computing power rather than any smart ideas; just by running YAFU, the factoring tool.
Possible to if $n\equiv_6-1$ and $n+2=b^k$ is power of some prime $b$, then $b\mid f(n)$.

Checked up to $n=10^5$ without exceptions.

gp-code:

nbk()=
{
 for(n=1, 10^5, f= n^(n+1)+(n+1)^(n+2);
  if(n%6==5,
   k= ispower(n+2, , &amp;b);
   if(k&amp;&amp;isprime(b),
    if(f%b==0,
     print("n = "n"    f("n")%"b" = "f%b"    b = "b"    k = "k)
     ,
     print("----     "n"    f("n")%"b" = "f%b"    b = "b"    k = "k);
     break()
    )
   )
  )
 )
};


Output:

n = 23    f(23)%5 = 0    b = 5    k = 2
n = 47    f(47)%7 = 0    b = 7    k = 2
n = 119    f(119)%11 = 0    b = 11    k = 2
n = 167    f(167)%13 = 0    b = 13    k = 2
n = 287    f(287)%17 = 0    b = 17    k = 2
n = 341    f(341)%7 = 0    b = 7    k = 3
n = 359    f(359)%19 = 0    b = 19    k = 2
n = 527    f(527)%23 = 0    b = 23    k = 2
n = 623    f(623)%5 = 0    b = 5    k = 4
n = 839    f(839)%29 = 0    b = 29    k = 2
n = 959    f(959)%31 = 0    b = 31    k = 2
n = 1367    f(1367)%37 = 0    b = 37    k = 2
n = 1679    f(1679)%41 = 0    b = 41    k = 2
n = 1847    f(1847)%43 = 0    b = 43    k = 2
n = 2195    f(2195)%13 = 0    b = 13    k = 3
n = 2207    f(2207)%47 = 0    b = 47    k = 2
n = 2399    f(2399)%7 = 0    b = 7    k = 4
n = 2807    f(2807)%53 = 0    b = 53    k = 2
n = 3479    f(3479)%59 = 0    b = 59    k = 2
n = 3719    f(3719)%61 = 0    b = 61    k = 2
n = 4487    f(4487)%67 = 0    b = 67    k = 2
n = 5039    f(5039)%71 = 0    b = 71    k = 2
n = 5327    f(5327)%73 = 0    b = 73    k = 2
n = 6239    f(6239)%79 = 0    b = 79    k = 2
n = 6857    f(6857)%19 = 0    b = 19    k = 3
n = 6887    f(6887)%83 = 0    b = 83    k = 2
n = 7919    f(7919)%89 = 0    b = 89    k = 2
n = 9407    f(9407)%97 = 0    b = 97    k = 2
n = 10199    f(10199)%101 = 0    b = 101    k = 2
n = 10607    f(10607)%103 = 0    b = 103    k = 2
n = 11447    f(11447)%107 = 0    b = 107    k = 2
n = 11879    f(11879)%109 = 0    b = 109    k = 2
n = 12767    f(12767)%113 = 0    b = 113    k = 2
n = 14639    f(14639)%11 = 0    b = 11    k = 4
n = 15623    f(15623)%5 = 0    b = 5    k = 6
n = 16127    f(16127)%127 = 0    b = 127    k = 2
n = 16805    f(16805)%7 = 0    b = 7    k = 5
n = 17159    f(17159)%131 = 0    b = 131    k = 2
n = 18767    f(18767)%137 = 0    b = 137    k = 2
n = 19319    f(19319)%139 = 0    b = 139    k = 2
n = 22199    f(22199)%149 = 0    b = 149    k = 2
n = 22799    f(22799)%151 = 0    b = 151    k = 2
n = 24647    f(24647)%157 = 0    b = 157    k = 2
n = 26567    f(26567)%163 = 0    b = 163    k = 2
n = 27887    f(27887)%167 = 0    b = 167    k = 2
n = 28559    f(28559)%13 = 0    b = 13    k = 4
n = 29789    f(29789)%31 = 0    b = 31    k = 3
n = 29927    f(29927)%173 = 0    b = 173    k = 2
n = 32039    f(32039)%179 = 0    b = 179    k = 2
n = 32759    f(32759)%181 = 0    b = 181    k = 2
n = 36479    f(36479)%191 = 0    b = 191    k = 2
n = 37247    f(37247)%193 = 0    b = 193    k = 2
n = 38807    f(38807)%197 = 0    b = 197    k = 2
n = 39599    f(39599)%199 = 0    b = 199    k = 2
n = 44519    f(44519)%211 = 0    b = 211    k = 2
n = 49727    f(49727)%223 = 0    b = 223    k = 2
n = 50651    f(50651)%37 = 0    b = 37    k = 3
n = 51527    f(51527)%227 = 0    b = 227    k = 2
n = 52439    f(52439)%229 = 0    b = 229    k = 2
n = 54287    f(54287)%233 = 0    b = 233    k = 2
n = 57119    f(57119)%239 = 0    b = 239    k = 2
n = 58079    f(58079)%241 = 0    b = 241    k = 2
n = 62999    f(62999)%251 = 0    b = 251    k = 2
n = 66047    f(66047)%257 = 0    b = 257    k = 2
n = 69167    f(69167)%263 = 0    b = 263    k = 2
n = 72359    f(72359)%269 = 0    b = 269    k = 2
n = 73439    f(73439)%271 = 0    b = 271    k = 2
n = 76727    f(76727)%277 = 0    b = 277    k = 2
n = 78959    f(78959)%281 = 0    b = 281    k = 2
n = 79505    f(79505)%43 = 0    b = 43    k = 3
n = 80087    f(80087)%283 = 0    b = 283    k = 2
n = 83519    f(83519)%17 = 0    b = 17    k = 4
n = 85847    f(85847)%293 = 0    b = 293    k = 2
n = 94247    f(94247)%307 = 0    b = 307    k = 2
n = 96719    f(96719)%311 = 0    b = 311    k = 2
n = 97967    f(97967)%313 = 0    b = 313    k = 2