## Monday, May 29, 2017

### [egxircfx] Factoring the numerators of the sum of prime reciprocals

A partial sum of the reciprocals of the prime numbers is a fraction whose denominator factors very easily (it is probably a primorial, though I have not proved it), but whose numerator is much more difficult to factor.  Below are some prime factorizations.  The three columns are an index, the nth prime, and the factorization, in the format of print(factorint) of Pari/GP.  (Not all the factorizations were computed with Pari/GP.)  The largest prime factor is elided for brevity.

1 2 matrix(0,2)
2 3 Mat([...])
3 5 Mat([...])
4 7 [13, 1; ...]
5 11 Mat([...])
6 13 Mat([...])
7 17 [19, 1; ...]
8 19 [37, 1; ...]
9 23 [43, 1; 163, 1; ...]
10 29 [269, 1; ...]
11 31 [61, 1; 4253, 1; ...]
12 37 [79, 1; 82913, 1; ...]
13 41 [2683561, 1; ...]
14 43 [47, 1; ...]
15 47 [1559, 1; ...]
16 53 [1701437, 1; ...]
17 59 [1087, 1; ...]
18 61 Mat([...])
19 67 Mat([...])
20 71 [653, 1; 24310877, 1; ...]
21 73 [79, 1; ...]
22 79 Mat([...])
23 83 [6991, 1; 17117, 1; 31727, 1; ...]
24 89 [128427649157, 1; ...]
25 97 [11887, 1; ...]
26 101 [2549, 1; ...]
27 103 [20831283974697087833, 1; ...]
28 107 [109, 1; 25013, 1; 45090757000504693, 1; ...]
29 109 [155963971, 1; ...]
30 113 [5507, 1; 67537, 1; 120585359, 1; 2153797249, 1; ...]
31 127 [4002997, 1; 164601157381, 1; ...]
32 131 [359111, 1; 2287463, 1; 901103850134867, 1; ...]
33 137 [181, 1; ...]
34 139 [1811, 1; 28393, 1; 22710555839, 1; ...]
35 149 [3923, 1; 574159, 1; 763111, 1; 24663715883, 1; ...]
36 151 [72279793, 1; 94335665844559, 1; ...]
37 157 [28547, 1; ...]
38 163 [4787, 1; 5521, 1; 72461, 1; 8380649, 1; 10408339, 1; 7832974103, 1; ...]
39 167 [278556893, 1; 3086934317, 1; ...]
40 173 [1907, 1; 10081481467, 1; ...]
41 179 [33390666307245425779, 1; ...]
42 181 [7219, 1; 220383301739, 1; 126474777435943, 1; ...]
43 191 [2382613, 1; 2446271, 1; 7448087, 1; 871590253, 1; 178268248033937558047, 1; ...]
44 193 [55721, 1; ...]
45 197 [331, 1; 75578568438239, 1; 38105656577309152682410376613803, 1; ...]
46 199 [397, 1; 1013, 1; 937077949469, 1; ...]
47 211 Mat([...])
48 223 [2882132344879945073347, 1; 45307571139926942991190133467, 1; ...]
49 227 [651126224977976987, 1; 318622749737882874703, 1; ...]
50 229 [85095942328560424348919161618392851191123181, 1; ...]
51 233 [2579, 1; 71532047297, 1; ...]
52 239 [32801, 1; 3646361, 1; 92115137111, 1; ...]
53 241 [1163, 1; 7313113087636909616856476683740381743, 1; ...]
54 251 [4457, 1; 4910826615619, 1; 1363395500165507803, 1; ...]
55 257 [339817, 1; 1513991, 1; 596996931741787, 1; 856758427621401709, 1; 2551502817340701509987573, 1; ...]
56 263 [353, 1; 27281214795174369762755731429871, 1; ...]
57 269 Mat([...])
58 271 Mat([...])
59 277 [769, 1; 17449, 1; 38853152699, 1; 25824790263806958200017, 1; ...]
60 281 [3931, 1; 4567121006246470847491526513, 1; ...]
61 283 [317, 1; 1483, 1; 1609, 1; 3083, 1; 15671, 1; 583879671521915894486003, 1; 40014655444673756284001412841, 1; ...]
62 293 [80309, 1; 43536091344561600390559851581507045347, 1; ...]
63 307 Mat([...])
64 311 [5147, 1; 358562916373373, 1; 1491945768116449, 1; ...]
65 313 [24197, 1; 180679, 1; 16666361, 1; 19487953638573362777, 1; 8677032348848311962094549, 1; ...]
66 317 [26881, 1; 1233681413, 1; 303909666003986239876228859266842827787705817105239453149, 1; ...]
67 331 ?
68 337 [4545229807, 1; 8458142681, 1; 143759487929629829, 1; ...]
69 347 [183919, 1; 2676456442899289, 1; 23977191281329122737, 1; 3403289789915363463733, 1; 141794625722542162388532383, 1; ...]
70 349 Mat([...])
71 353 [1453, 1; 831263358975239, 1; ...]
72 359 [10774822439848219544475069839689, 1; 122797736200334999541587307307277, 1; 2437316904825953681023538533923768491, 1; ...]
73 367 Mat([...])
74 373 [36637, 1; ...]
75 379 [114599, 1; 2560985953, 1; 245244287749171, 1; ...]
76 383 [24509, 1; 10904541719992241, 1; 180014284879715834547510101249288391580157, 1; ...]
77 389 [124513, 1; 3716098294674500690132872478501721045666345109541, 1; ...]
78 397 [1237, 1; 40087, 1; 193883, 1; 71681682691, 1; 8335042497553, 1; 6407545381506250342667407731823103, 1; 1632270826911851865940901782780510543, 1; ...]
79 401 [15053, 1; 39387829631, 1; 17246820026707, 1; 35482216430978507, 1; ...]
80 409 [104287094536357, 1; 315148692435993889067971, 1; ...]
81 419 [2671, 1; ...]
82 421 [1661985439636800100353810329, 1; ...]
83 431 ?
84 433 [3752943149479, 1; ...]
85 439 ?
86 443 [1049, 1; ...]
87 449 ?