divisor and antidivisor
thus, "repunit" and "antirepunit"
the "repunit" with "index" n in base b is b^(n-1)+b^(n-2)+b^(n-3)+...+b^2+b+1
the "antirepunit" with "index" n in base b is b^(n-1)-b^(n-2)-b^(n-3)-...-b^2-b-1
(although the "repunit" with "index" n in base b has length n in base b, but the "antirepunit" with "index" n in base b only has length n-1 in base b)
this is a list of indices n<=2000 such that the "repunit"/"antirepunit" with "index" n in base b is prime:
2, repunit: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279,
2, antirepunit:
3, repunit: 3, 7, 13, 71, 103, 541, 1091, 1367, 1627,
3, antirepunit: 2, 3, 5, 17, 33, 65,
4, repunit: 2,
4, antirepunit: 2, 3, 4, 6, 7, 9, 10, 12, 16, 22, 31, 40, 51, 64, 84, 96, 100, 157, 174, 351, 855, 1309, 1770,
5, repunit: 3, 7, 11, 13, 47, 127, 149, 181, 619, 929,
5, antirepunit: 3, 7, 9, 63, 121, 187, 415, 765,
6, repunit: 2, 3, 7, 29, 71, 127, 271, 509, 1049,
6, antirepunit: 2, 3, 4, 6, 16, 26, 30, 74, 91, 140, 195, 243, 940,
7, repunit: 5, 13, 131, 149, 1699,
7, antirepunit: 3, 7, 57, 147,
8, repunit: 3,
8, antirepunit: 2, 4, 5, 6, 24, 74, 1034, 1170, 1901,
9, repunit:
9, antirepunit: 3, 5, 7, 17, 381, 387,
10, repunit: 2, 19, 23, 317, 1031,
10, antirepunit: 3, 15, 18, 36,
11, repunit: 17, 19, 73, 139, 907, 1907,
11, antirepunit: 3, 5, 19, 31, 193,
12, repunit: 2, 3, 5, 19, 97, 109, 317, 353, 701,
12, antirepunit: 2, 3, 4, 22, 40, 100, 158, 262, 1719,
13, repunit: 5, 7, 137, 283, 883, 991, 1021, 1193,
13, antirepunit: 565,
14, repunit: 3, 7, 19, 31, 41,
14, antirepunit: 2, 3, 5, 6, 17, 19, 30, 43, 83, 88, 147, 167, 591, 638, 942, 1249, 1305, 1670,
15, repunit: 3, 43, 73, 487,
15, antirepunit: 11, 181,
16, repunit: 2,
16, antirepunit: 3, 4, 10, 45, 84, 1024,
17, repunit: 3, 5, 7, 11, 47, 71, 419,
17, antirepunit: 3, 5, 7, 15, 29, 87, 319, 383, 563, 825, 1059, 1227,
18, repunit: 2,
18, antirepunit: 2, 5, 6, 9, 10, 16, 32, 68, 80, 89, 125, 266, 389, 1390,
19, repunit: 19, 31, 47, 59, 61, 107, 337, 1061,
19, antirepunit: 15, 55, 331,
20, repunit: 3, 11, 17, 1487,
20, antirepunit: 2, 3, 5, 6, 12, 42, 48, 75, 77, 82, 670, 852,
21, repunit: 3, 11, 17, 43, 271,
21, antirepunit: 3, 99, 103, 415,
22, repunit: 2, 5, 79, 101, 359, 857,
22, antirepunit: 3, 4, 447, 504,
23, repunit: 5,
23, antirepunit: 9, 45, 265, 345, 1857,
24, repunit: 3, 5, 19, 53, 71, 653, 661,
24, antirepunit: 2, 5, 6, 9, 27, 701, 816, 1105,
25, repunit:
25, antirepunit: 3, 9, 1351,
26, repunit: 7, 43, 347,
26, antirepunit: 6, 94, 334, 584, 756, 1515, 1634,
27, repunit: 3,
27, antirepunit: 3, 5, 13, 33, 85, 109, 231, 751,
28, repunit: 2, 5, 17, 457, 1423,
28, antirepunit: 4, 6, 10, 13, 22, 72, 1752,
29, repunit: 5, 151,
29, antirepunit: 3, 803, 1403,
30, repunit: 2, 5, 11, 163, 569, 1789,
30, antirepunit: 2, 7, 8, 51, 69, 87, 372, 634, 1636,
31, repunit: 7, 17, 31,
31, antirepunit: 3, 51, 163, 261, 385, 483, 751, 1123,
32, repunit:
32, antirepunit: 2, 3, 6, 24, 28, 54, 176, 349, 1009,
33, repunit: 3, 197,
33, antirepunit: 253, 1881,
34, repunit: 13, 1493,
34, antirepunit: 4, 25, 172, 994, 1843,
35, repunit: 313, 1297,
35, antirepunit: 21, 23, 1881,
36, repunit: 2,
36, antirepunit: 3, 15, 49, 101,
37, repunit: 13, 71, 181, 251, 463, 521,
37, antirepunit: 7, 19, 175, 187,
38, repunit: 3, 7, 401, 449,
38, antirepunit: 2, 5, 8, 36,
39, repunit: 349, 631,
39, antirepunit: 3, 7, 9, 13, 163, 355, 663, 935,
40, repunit: 2, 5, 7, 19, 23, 29, 541, 751, 1277,
40, antirepunit: 3, 6, 13, 27, 39, 51, 166, 372, 705, 720, 1332, 1417, 1542,
41, repunit: 3, 83, 269, 409, 1759,
41, antirepunit: 5, 9, 15, 19, 295,
42, repunit: 2, 1319,
42, antirepunit: 2, 3, 6, 7, 116, 141, 191, 645, 1897,
43, repunit: 5, 13,
43, antirepunit: 25,
44, repunit: 5, 31, 167,
44, antirepunit: 2, 4, 23, 92, 186, 662, 1107,
45, repunit: 19, 53, 167,
45, antirepunit: 3, 5, 15,
46, repunit: 2, 7, 19, 67, 211, 433,
46, antirepunit: 3, 30, 84,
47, repunit: 127,
47, antirepunit: 3, 253, 283,
48, repunit: 19, 269, 349, 383, 1303,
48, antirepunit: 2, 5, 16, 57, 65, 285, 466, 609, 1357,
49, repunit:
49, antirepunit: 3, 9, 21, 925, 1065, 1713,
50, repunit: 3, 5, 127, 139, 347, 661,
50, antirepunit: 4, 5, 6, 27, 57, 151, 204, 665, 734,
51, repunit:
51, antirepunit: 3, 5, 21, 153, 215,
52, repunit: 2, 103, 257,
52, antirepunit: 18, 69, 151, 558, 1089, 1686,
53, repunit: 11, 31, 41, 1571,
53, antirepunit: 25, 85, 221,
54, repunit: 3, 389,
54, antirepunit: 2, 3, 4, 14, 38, 43, 58, 278, 310, 1459, 1567,
55, repunit: 17, 41, 47, 151, 839,
55, antirepunit: 3, 79, 1585,
56, repunit: 7, 157,
56, antirepunit: 3, 4, 5, 11, 20, 29, 65, 161, 694, 1771, 1904,
57, repunit: 3, 17, 109, 151, 211, 661,
57, antirepunit: 3, 29, 45, 63, 91, 459, 703, 1519,
58, repunit: 2, 41,
58, antirepunit: 4, 6, 18, 132, 208, 418,
59, repunit: 3, 13, 479,
59, antirepunit: 5, 235, 457, 549,
60, repunit: 2, 7, 11, 53, 173,
60, antirepunit: 2, 3, 5, 23, 25, 51, 57, 67, 993, 1032, 1636,
61, repunit: 7, 37, 107, 769,
61, antirepunit: 3, 9, 31, 259,
62, repunit: 3, 5, 17, 47, 163, 173, 757,
62, antirepunit: 2, 5, 6, 7, 45, 56, 460, 970,
63, repunit: 5,
63, antirepunit: 5, 37, 981,
64, repunit:
64, antirepunit: 12, 357, 1168, 1308,
65, repunit: 19, 29, 631,
65, antirepunit: 3, 11, 115, 291,
66, repunit: 2, 3, 7, 19,
66, antirepunit: 3, 8, 648, 818,
67, repunit: 19, 367, 1487,
67, antirepunit: 3, 33,
68, repunit: 5, 7, 107, 149,
68, antirepunit: 2, 6, 9, 18, 20, 21, 45, 68, 81, 98, 133, 642,
69, repunit: 3, 61,
69, antirepunit: 3, 7, 25, 241, 531,
70, repunit: 2, 29, 59, 541, 761, 1013,
70, antirepunit: 556, 1075,
71, repunit: 3, 31, 41, 157, 1583,
71, antirepunit: 3, 5, 61,
72, repunit: 2, 7, 13, 109, 227,
72, antirepunit: 2, 6, 7, 127, 476, 1801,
73, repunit: 5, 7,
73, antirepunit: 9, 661,