Prime factors of numbers of the form $n^{n^n}+n^n+1$

Question

Here :

Prime candidates of the form $n^{(n^n)}+n^n+1$?

I asked for the prime factors of $f(12)$ and $f(60)$, where $$f(n)=n^{(n^n)}+n^n+1$$

I would like to accelarate the search of the prime factors of $f(n)$.

Are there special properties considerably restricting the possible prime factors of $f(n)$ ?

The only numbers I factored completely so far, are $f(1),f(2),f(3)$ and $f(4)$. For those, who are interested, here are the factoriations :

https://factordb.com/index.php?query=n%5En%5En%2Bn%5En%2B1

Answer 1

Small prime factors of the expression (to 100k), for use in testing any hypothesis.

Observations: Multiples of $6$ have some kind of restricted options going on. Numbers $\{1,2,4\}\bmod 6$ have the expression divisible by $3$, others not. If $n{+}2$ is an odd prime or prime power, that prime divides the expression. If $2n{-}1$ is prime, that is often a factor.

n   factors                                                 
1   3                                                   
2   3   7                                               
3   5                                                   
4   3   19  3457    35149                                       
5   7   109 11213   63487                                       
6   11                                                  
7   3   13  67  10477                                       
8   3   19  37  73  109 433 38737                           
9   11                                                  
10  3   19                                              
11  13  5737                                                
12                                                      
13  3                                                   
14  3   23  43  61  211 463 547                         
15  17  29                                              
16  3   11  131 457 2731    80611                               
17  5   19  661 9127                                        
18  1319                                                    
19  3   17  37                                          
20  3   13  31  67  127 421 3001    12277   71161                   
21  23  83                                              
22  3   31  43  383 5717                                    
23  5   7949                                                
24  4549    55807                                               
25  3   17  19  139 233 91841                               
26  3   7   19  31  37  79  313 547 859 15607               
27  29  53                                              
28  3   19  53  1667    16183   31159                               
29  31                                                  
30  59                                                  
31  3   7   61                                          
32  3   7   13  61  97  151 193 241 331 673 1321    23041       
33  41                                                  
34  3   67  29501                                           
35  37  91243                                               
36  911                                                 
37  3   5   11  47                                      
38  3   7   59  67  1483    3079    25537                           
39  41  347                                             
40  3   1451                                                
41  13  43                                              
42  83                                                  
43  3   5   421 797 1559    8803                                
44  3   7   67  283 631 1753    2137    28447   38611   46993               
45  47  179 32009                                           
46  3   17  173                                         
47  7                                                   
48  23                                                  
49  3                                                   
50  3   19  31  43  601 1051    1201    2551    44371                   
51  53  101 30593                                           
52  3   31  151 27283                                       
53  7074407                                                 
54  17  103 107                                         
55  3   109                                             
56  3   13  31  43  73  79  103 3361    49393   70921               
57  5   59  227 239 719                                 
58  3   83  797                                         
59  61                                                  
60                                                      
61  3   41                                              
62  3   13  71  97  179 1303    1489    3907    8929                    
63  5   317 3617                                            
64  3   43  3253                                            
65  67                                                  
66  131                                                 
67  3   163                                             
68  3   7   13  19  31  53  103 109 199 307 613 4007    23869   38149
69  71                                                  
70  3   19  89  139                                     
71  11  73                                              
72  1451    2389                                                
73  3   7   8641                                            
74  3   7   13  17  61  71  223 1801    3191                    
75  23  149 5419                                            
76  3   2267                                                
77  5   17  41  79  101 11867                               
78  16981                                                   
79  3   37  157                                         
80  3   7   13  31  43  61  73  241 6481    27697   50221   60937       
81  83  12451                                               
82  3   163                                             
83  5   47  97  293                                     
84  53                                                  
85  3   79  233                                         
86  3   7   1069    2437                                        
87  89  173 491                                         
88  3   59  653 18493                                       
89  7   439 8689                                            
90  179                                                 
91  3   181 8429                                            
92  3   43  139 199 277 967 1381    2791                        
93  29                                                  
94  3                                                   
95  97  3089    3943    27127                                       
96  53  211                                             
97  3   5   13                                          
98  3   31  43  313 3169                                    
99  101 197 48413                                           
100 3   67