[kgkiymbk] Factoring subpermutations

A000522 (total number of arrangements) grows faster than exponentially, faster than factorial.

Here are some factorizations up to 73 with the largest factor is elided for compactness.

? f(n)=sum(i=0,n,binomial(n,i)*i!)

? for(q=0,1000,print(q," ",factor(f(q))))
0 matrix(0,2)
1 Mat([2, 1])
2 Mat([5, 1])
3 Mat([2, 4])
4 [5, 1
5 [2, 1
6 [19, 1
7 [2, 2; 5, 2
8 [127, 1
9 [2, 1; 5, 1
10 [13, 1; 71, 1
11 [2, 3; 431, 1
12 [5, 1; 13, 2
13 [2, 1; 16189, 1
14 [5, 1; 19, 1; 23, 1; 31, 1; 643, 1
15 [2, 2; 3917, 1
16 [37, 1; 137, 1; 751, 1; 1097, 1
17 [2, 1; 5, 1; 13, 1; 491, 1; 11863, 1
18 [23, 1; 1827593, 1
19 [2, 5; 5, 3; 31, 1; 43, 1; 29191, 1
20 [389, 1; 1341737, 1
21 [2, 1; 271, 1; 305761, 1
22 [5, 1; 131, 1; 47741, 1; 2187851, 1
23 [2, 2; 13, 1; 29, 2; 2861, 1; 1734071, 1
24 [5, 1
25 [2, 1; 13, 1; 19, 1; 3463, 1; 41231, 1; 10997999, 1
26 [97, 1
27 [2, 3; 5, 1; 7218853, 1
28 [251, 1; 18301, 1
29 [2, 1; 5, 1; 24136781, 1; 278078117, 1
30 [13, 1; 142131853, 1
31 [2, 2; 197, 1; 1367, 1; 893119, 1; 41945543, 1
32 [5, 2
33 [2, 1; 19, 1; 131, 1; 181873927531, 1
34 [5, 1; 37, 1; 839, 1; 3015163, 1
35 [2, 4; 21683, 1
36 [13, 1; 37, 1; 4661647357, 1; 154259933251, 1
37 [2, 1; 5, 1; 23, 1; 41, 1; 313, 1; 877, 1; 301127, 1; 1049681, 1
38 [13, 1; 287399003, 1; 189098406503685881, 1
39 [2, 2; 5, 1; 839, 1; 26014013, 1; 101888516607554881, 1
40 [2393, 1; 185807005001824729, 1
41 [2, 1; 23, 1; 59, 1; 12721, 1; 8727704012195599, 1
42 [5, 1; 19876613, 1; 16005034157141593141, 1
43 [2, 3; 13, 1; 5683, 1; 18749, 1; 47791, 1; 109721, 1; 955600776733, 1
44 [5, 2; 19, 1; 1163, 1; 53492475499, 1; 1548419650850851, 1
45 [2, 1; 31, 1; 53564218996596383, 1
46 [100120183, 1; 740579588201, 1
47 [2, 2; 5, 1; 173750833079, 1; 166246982779321, 1; 21591978383440183, 1
48 [964832846111, 1
49 [2, 1; 5, 1; 13, 1; 71, 1; 28723, 1; 1677251, 1; 103045931200679715862687, 1
50 [31, 1; 113, 1; 320693, 1
51 [2, 6; 13, 1; 107, 1; 223619818437724981434781, 1
52 [5, 1; 19, 1; 29, 1; 56263, 1; 84155537, 1; 30824778703, 1; 750889242216399697, 1
53 [2, 1; 37, 1; 169313, 1; 637337, 1; 12177765169, 1; 77088822797611, 1; 2313143852634061, 1
54 [5, 1; 27583, 1; 343327, 1; 1286693, 1; 582142854376447309163, 1
55 [2, 2; 1222975349, 1; 724886917038388850413, 1
56 [13, 1; 191672707, 1; 552371951, 1; 127152839700163, 1; 59071803788421013, 1
57 [2, 1; 5, 2; 193, 1; 1601573069597, 1; 38772475446796029217, 1
58 [211, 1; 8317, 1; 112269265983913, 1
59 [2, 3; 5, 1; 5376421, 1; 146425207, 1; 12483887887, 1; 26206294154055714550115731, 1
60 [23, 1; 8353577849059, 1; 215237107165294577972284698523, 1
61 [2, 1; 4657963, 1; 8841446055570861853, 1; 6321047070124191664607, 1
62 [5, 1; 13, 1; 43, 1; 193, 1; 197647, 1; 36252479, 1; 519248173108214565942109761341, 1
63 [2, 2; 19, 1; 23276347, 1; 56964720094654586464041967696120542857, 1
64 [5, 1; 13, 1; 23, 2; 9119286969899299, 1
65 [2, 1; 409, 1; 23819, 1; 319937, 1; 1629547, 1; 527431938570417763, 1
66 [331, 1; 1237, 1; 431803, 1; 396700951, 1; 524384294304817, 1
67 [2, 4; 5, 1; 6011, 1; 2568469, 1; 9803095242420956135137, 1
68 [11555812522107631, 1; 560944051850730826043, 1; 919623579090658367059403267, 1
69 [2, 1; 5, 2; 13, 1; 149, 1; 353, 1; 14009, 1; 232853, 1; 135522019033, 1; 1234249378957019648774949313, 1
70 [1510406442204189241, 1; 4044638573158080895672864223, 1
71 [2, 2; 19, 1; 37, 1; 23663, 1; 124526248151, 1
72 [5, 1; 5717691113, 1; 30231911414112917, 1; 146852904002052901, 1; 7395106465738090439, 1
73 [2, 1; 37, 2; 113891, 1; 5269276346332266349, 1; 31513747150137530028767, 1
*** factor: Warning: MPQS: factoring this number will take many hours:
N = 2166690334049728637398152472258997933316264469602300239945569693840859184157823454814218981879047682695483.
*** factor: Warning: MPQS: Gauss elimination will require more than
128MBy of memory.
Killed

Update: 74 through 95.