Off by one

for(a=1,50,print(a," ",factor(a^a*a+a+1)))

1 Mat([3, 1])
2 Mat([11, 1])
3 [5, 1; 17, 1]
4 [3, 1; 7, 3]

5	7, 2; 11, 1; ...
6	271, 1; ...
7	3, 1; 19, 2; ...
8	17, 1; 1361, 1; ...
9	11, 1; 19, 1; ...
10	3, 1; 37, 2; ...
11	13, 1; 36529, 1; ...
12	5, 1; 71, 1; ...
13	3, 1; 19, 1; 61, 2; 461, 1; 2713, 1; ...
14	277, 1; ...
15	17, 1; 63857, 1; ...
16	3, 1; 7, 2; 13, 2; ...
17	11, 1; 19, 1; ...
18	201000229, 1; ...
19	3, 1; 127, 2; 167, 1; 130841, 1; 936283, 1; ...
20	41, 1; 53549, 1; 244813103, 1; ...
21	19, 1; 23, 1; 43, 1; 461, 1; ...
22	3, 1; 13, 4; 19, 2; 239, 1; 3019, 1; 28439, 1; ...
23	5, 1; ...
24	7, 1; 31, 1; 367, 1; 3697, 1; 4019, 1; 4421, 1; ...
25	3, 1; 7, 3; 31, 2; 47, 1; 283, 1; 164485232117, 1; ...
26	1151, 1; 1949, 1; 2420321, 1; 2994661, 1; 192604459, 1; ...
27	29, 1; 2087, 1; 20693, 1; 127703, 1; 1496783, 1; ...
28	3, 1; 23, 1; 271, 2; 7349, 1; 7031023, 1; 16386187, 1; ...
29	31, 1; 59, 1; 179, 1; 287356731133, 1; ...
30	23, 1; 251, 1; 416598973451, 1; 614391925556407, 1; ...
31	3, 1; 211, 1; 331, 2; 31883, 1; 71147, 1; 278867, 1; 21162014243, 1; ...
32	5, 1; 379, 1; 183022069, 1; ...
33	67, 1; ...
34	3, 1; 113, 1; 397, 2; 90620892826724569279, 1; ...
35	37, 1; 11497, 1; 1534957, 1; 9783807889, 1; 69375842508001, 1; ...
36	73, 1; 79, 1; 11813, 1; 4369097, 1; 22909204931549, 1; 169228131318583, 1; ...
37	3, 1; 7, 2; 23, 1; 67, 2; 179, 1; 17851, 1; ...
38	271, 1; 109398304199, 1; ...
39	41, 1; 641, 1; 551028832864379, 1; ...
40	3, 1; 23, 1; 547, 2; 163193, 1; 103821313, 1; ...
41	43, 1; 83, 1; 249341, 1; 292541, 1; ...
42	7673, 1; ...
43	3, 1; 5, 1; 31, 1; 47, 1; 631, 2; 390050231523401657587, 1; ...
44	71, 1; 89, 1; 1677045346368068427777407257, 1; ...
45	47, 1; 9051947, 1; 491229703, 1; 14128833035515081731187, 1; ...
46	3, 1; 7, 3; 103, 2; 1439, 1; 6113, 1; 1417134681070517, 1; 1211221369572027119, 1; ...
47	7, 1; 208498453, 1; 62768966677, 1; ...
48	43, 1; 97, 1; 1031295286537, 1; ...
49	3, 1; 19, 2; 43, 2; 8737, 1; 153409, 1; 4161881, 1; 47557729, 1; 1231572632971, 1; 8245918385603, 1; ...
50	130657613, 1; 96872622227, 1; 218906875337, 1; 28924899119393713, 1; ...

51	53, 1; 2629867675577, 1; 569443060925776280407727439893, 1; ...
52	3, 1; 5, 1; 23, 1; 919, 2; ...
53	107, 1; 1999, 1; 737641, 1; 930412261043, 1; 94130302460041, 1; 3192743294266334761, 1; ...
54	96017, 1; 260419, 1; 8196173, 1; ...
55	3, 1; 13, 2; 79, 2; 643, 1; 971, 1; 7283, 1; 7457, 1; 76423, 1; 12409492464988175419299322668289387, 1; ...
56	113, 1; 241, 1; 353, 1; 953, 1; 469411, 1; 361014333137116391917, 1; 3628516781012996187539369, 1; ...
57	59, 1; 100386889, 1; 6321543675100916653, 1; 341493650649284541536291, 1; ...
58	3, 1; 7, 2; 163, 2; ...
59	53, 1; 61, 1; 349, 1; 9245158309, 1; 120544548211630908902423, 1; 78160655788202033012841388117831, 1; ...
60	11, 1; 43, 1; 47, 1; 199, 1; 1171, 1; 760719804161, 1; 1964835377513, 1; 1270629830046613, 1; ...
61	3, 1; 13, 2; 97, 2; 181, 1; 5947512368423, 1; 593123889824783, 1; 118344653044407041528153987, 1; ...
62	13355959, 1; 164496841, 1; 10292091945465437025733, 1; 74699533901627257475568411525705151, 1; ...
63	5, 2; 41, 1; 853, 1; 90174101, 1; 25627454210533711458438181, 1; 5919051349510792163062037098889, 1; ...
64	3, 1; 11, 1; 19, 2; 29, 1; 73, 2; 63223491756531089, 1; ...
65	67, 1; 131, 1; 75931, 1; 161406967771214788618904596583487255863, 1; ...
66	7, 1; 13229, 1; 175621, 1; 60705385096830613, 1; 7569220110914916487333209391763, 1; ...
67	3, 1; 7, 5; 31, 2; 166807, 1; 125620306831, 1; 12633727094391709793406313, 1; 1297214578001286301761672282562841, 1; ...
68	137, 1; 4457, 1; ...
69	71, 1; 97, 1; 139, 1; 32051, 1; ...
70	3, 1; 1657, 2; 23239339, 1; 3197199066319589985643186083781752161, 1; ...
71	73, 1; 8009, 1; 11753737, 1; 81534544530059862557486363, 1; 168289332683800594045130452406607652729, 1; ...
72	5, 3; 29987982540984257, 1; 1181241087252894645457638859178483, 1; ...
73	3, 1; 11, 2; 67, 1; 1801, 2; 65677, 1; 342044282612209, 1; ...
74	?
75	311, 1; 557, 1; 1871, 1; 4357651, 1; 48515144351, 1; ...
76	?
77	79, 1; 131, 1; 4703212510091, 1; 761643630403488538123655879, 1; ...
78	?