Another sequence of note

In addition to the pentatrigesimal sequences I looked at last year — based upon using all numbers and latters except O — I have studied another sequence of numbers which I will tabulate below. Of the first 104 terms which I am tabulating, one is not yet known and is shaded in grey background with white text.

n	p
2	2
3	71
4	0
5	11
6	29
7	131
8	0
9	0
10	23
11	73
12	97
13	137
14	41
15	43
16	0
17	419
18	25667
19	59
20	1487
21	156217
22	79
23	3181
24	53
25	0
26	347
27	0
28	457
29	151
30	163
31	5581
32	0
33	197
34	1493
35	313
36	0
37	251
38	401
39	349
40	751
41	83
42	1319
43	6277
44	167
45	3319
46	67
47	18013
48	383
49	0
50	6521
51	4229
52	257
53	1571
54	389
55	839
56	157
57	16963
58	2333
59	479
60	173
61	37
62	757
63	3067
64	0
65	375017
66	19973
67	367
68	2767
69	2371
70	761
71	1583
72	227
73	110603
74	191
75	739
76	439
77	15361
78	1949
79	659
80	>399989
81	0
82	7607
83	2713
84	3917
85	2111
86	113
87	121487
88	577
89	571
90	5209
91	4421
92	13001
93	4903
94	170371
95	523
96	3343
97	1693
98	2801
99	5563
100	0
101	677
102	673
103	1549
104	263
105	4783

This sequence is — in essence — the inverse of OEIS sequence A066180. The nth term of this sequence is the smallest prime for which n is the smallest base yielding a generalised repunit prime. Alternatively, the nth term is defined as the first prime number yielding n in sequence A066180.

For bases that are perfect powers, generalised repunits can be factored algebraically and the sequence has the value 0. For base 65 — until the recent discovery of the probable prime 65³⁷⁵⁰¹⁷-1/64 — and base 80, no known generalised repunit prime exists or existed without a smaller base yielding a generalised repunit prime, as can be seen from the table below:

base	primes	Smaller bases where Rp is prime
65	19	2, 10, 11, 12, 14, 19, 24, 40, 45, 46, 48
	29	6, 40
	631	39
80	3	2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62
	7	2, 3, 5, 6, 13, 14, 26, 31, 38, 40, 46, 56, 60, 61, 66, 68, 72, 73

At present I am not sure over what range bases 65 and 80 have been checked, although data for adjacent bases from Henri and Renauld Lifchitz suggests they have probably been checked up to around four hundred thousand without additional generalised repunit primes being discovered.

As a last word, it might be noted that, of the first three hundred primes (up to 1987), eighty-three do not appear in this sequence at all, viz:

3	5	7	13	17	19	23	31	47	61	73	89
97	101	103	107	109	127	131	137	139	149	181	211
269	271	283	317	337	353	359	409	433	449	463	487
509	521	541	569	587	593	607	619	631	653	661	701
757	769	821	857	883	907	929	971	991	1013	1021	1031
1049	1061	1069	1087	1091	1151	1181	1193	1277	1279	1297	1303
1367	1409	1423	1487	1627	1699	1721	1759	1789	1861	1907

For these primes, the first base yielding a generalised repunit prime is also the first such base for a smaller prime. For instance:

1. for 3, 5, and 7, base 2 is the first such base, but 2²-1 is also prime
2. for 103, 541, 1091 and 1367, base 3 is the first base yielding a prime but (3⁷¹-1)/2 is also prime
3. for 317 and 1031, base 10 is the first base yielding a prime, but (10²³-1)/9 is also prime