An alternative view of primes in centuries

The OEIS sequences at A038822 and A186311 have long been familiar to me ever since I first studied primes as a high school student reading the old Mathematics Around Us: Skills and Applications book from my primary school years. As a teenager, I would attempt to factor numbers beyond the prime table my late uncle gave me and memorises the primes up to 2,000 — something which I like to practice with my mobile on buses and even to quiz others with no interest in primes about.

When I studied primes in the second millennium, I notices very early on that there was a discrepancy between centuries in the number of possible primes, although at the time I naturally could not recognise the exact extent of the actual limits found in sequence A186311. I noticed that

• every third century had 28 numbers not divisible by 2, 3, or 5 —
• I knew elementary divisibility tests for the prime numbers 2, 3, and 5 as a child, so I knew immediately numbers which passed those tests could not be prime
• the other two centuries in each set of 3 had only 26 numbers not divisible by 2, 3, or 5
• that centuries with 28 numbers not divisible by 2, 3, or 5 began with a number of the form 300k+100
• that centuries with 26 numbers not divisible by 2, 3, or 5 began with a number of the form 300k or 300k+200

As I memorised primes, I also memorised the factors of composite numbers not divisible by 2, 3, or 5 — all of them up to 2,000 and sporadically up to 10,000 and even beyond in some special cases. Although I was not aware of it until recently, I early on noticed clusters with unusually many or few numbers whose factorisations required memorisation — in layman’s terms, composite numbers not divisible by 2, 3, or 5.

It is only recently, when I have come to study prime-rich centuries beyond the “eight-digit gap” (in layman’s terms, numbers greater than 839,296,300) that I have attempted to actually sequence the number of composite numbers not divisible by 2, 3 or 5 in each century. The sequence of number of composite numbers not divisible by 2, 3, or 5 between 100k and 100k+99 begins thus:

3, 7, 10, 10, 11, 12, 10, 14, 11, 12, 12, 14, 11, 17, 9, 14, 13, 14, 14, 15, 12, 16, 13, 11, 16, 17, 11, 12, 16, 15, 14, 18, 15, 11, 17, 12, 13, 16, 15, 15, 13, 17, 10, 19, 15, 14, 16, 14, 18, 13, 14, 15, 18, 16, 13, 15, 14, 16, 12, 19, 14, 17, 13, 11, 20, 15, 16, 16, 14, 13, 19, 16, 15, 19, 15, 11, 16, 16, 16, 18, 15, 16, 14, 17, 18, 16, 13, 15, 15, 17, 15, 16, 15, 15, 13, 19, 13, 17, 14, 17, 17, 14, 16, 16, 16, 18, 16, 15, 16, 18, 16, 15, 18, 16, 15, 19, 18, 17, 16, 13, 17, 17, 14, 17, 15, 14, 14, 18, 17, 14, 17, 14, 17, 18, 15, 18, 16, 14, 17, 17, 16, 18, 21, 17, 14, 16, 16, 12, 16, 18, 17, 16, 14, 14, 17, 18, 13, 16, 17, 16, 16, 17, 18, 19, 14, 19, 15, 19, 17, 16, 15, 17, 19, 14, 13, 18, 18, 16, 20, 12, 16, 17, 15, 15, 17, 16, 20, 17, 21, 18, 17, 19, 15, 19, 10, 15, 21, 16, 16, 16, 17, 13, 19, 14, 18, 18, 19, 14, 19, 17, 15, 15, 19, 15, 18, 13, 17, 17, 15, 18, 15, 14, 17, 19, 18, 17, 16, 15, 18, 19, 10, 18, 18, 17, 20, 16, 14, 17, 15, 17, 13, 16, 19, 18, 18, 18, 17, 20, 18, 15, 20, 15, 19, 16, 16, 18, 16, 16, 18, 16, 18, 15, 17, 16, 16, 20, 16, 15, 16, 16, 16, 20, 16, 21, 18, 18, 17, 14, 17, 15, 17, 19, 17, 21, 15, 14, 14, 16, 17, 20, 17, 15, 18, 15, 17, 18, 18, 19, 18, 19, 17, 16, 17, 17, 19, 17, 17, 20, 13, 18, 18, 14, 15, 16, 22, 16, 18, 16, 19, 22, 13, 17, 19, 13, 15, 17, 17, 16, 20, 13, 16, 18, 20, 14, 18, 14, 16, 16, 16, 18, 22, 17, 15, 17, 16, 16, 17, 17, 16, 21, 16, 17, 19, 17, 17, 15, 22, 19, 16, 16, 16, 20, 16, 17, 19, 15, 16, 16, 17, 15, 18, 18, 19, 16, 18, 11, 18, 21, 16, 19, 20, 17, 18, 16, 20, 21, 14, 15, 19, 16, 18, 16, 17, 14, 21, 17, 18, 17, 14, 18, 19, 15, 18, 23, 16, 16, 21, 19, 13, 20, 17, 15, 15, 18, 19, 18, 13, 19, 17, 14, 16, 18, 16, 16, 13, 19, 17, 15, 18, 16, 19, 19, 18, 21, 17, 18, 18, 15, 21, 17, 16, 16, 15, 20, 20, 17, 15, 16, 19, 16, 20, 17, 19, 16, 20, 16, 17, 21, 15, 19, 19, 16, 19, 19, 16, 17, 17, 18, 15, 22, 19, 16, 19, 16, 18, 16, 17, 15, 20, 15, 19, 18, 19, 18, 16, 17, 17, 18, 15, 20, 18, 14, 17, 19, 15, 18, 18, 17, 16, 18, 16, 16, 19, 16, 19, 16, 21, 18, 19, 17, 19, 18, 17, 18, 14, 15, 16, 20, 15, 17, 20, 16, 17, 19, 22, 14, 20, 17, 17, 17, 17, 15, 19, 19, 18, 19, 14, 18, 17, 18, 18, 19, 20, 17, 15, 15, 17, 18, 20, 17, 19, 19, 16, 18, 20, 19, 14, 18, 14, 18, 17, 17, 19, 19, 15, 16, 18, 16, 17, 13, 18, 17, 16, 16, 22, 18, 17, 16, 17, 18, 18, 15, 18, 18, 16, 19, 18, 17, 23, 16, 14, 16, 18, 16, 15, 21, 14, 16, 23, 19, 17, 18, 17, 18, 21, 21, 13, 17, 19, 17, 19, 20, 18, 20, 16, 17, 15, 19, 19, 18, 16, 17, 20, 19, 17, 19, 17, 18, 20, 15, 19, 20, 20, 13, 17, 17, 13, 17, 18, 19, 19, 18, 19, 20, 18, 20, 17, 19, 18, 20, 16, 14, 20, 18, 17, 16, 16, 16, 18, 17, 18, 19, 22, 17, 19, 16, 19, 18, 17, 17, 19, 15, 16, 21, 13, 16, 21, 15, 17, 16, 19, 20, 19, 20, 17, 19, 18, 16, 19, 19, 18, 18, 18, 18, 14, 23, 20, 21, 16, 19, 18, 14, 16, 19, 17, 17, 19, 19, 17, 14, 19, 17, 19, 13, 15, 19, 20, 18, 16, 15, 17, 20, 15, 18, 20, 20, 15, 19, 18, 14, 18, 20, 21, 18, 18, 16, 17, 19, 17, 21, 17, 16, 18, 17, 18, 17, 20, 14, 17, 20, 18, 20, 16, 16, 22, 14, 16, 16, 21, 16, 20, 19, 16, 21, 17, 17, 19, 19, 17, 20, 17, 19, 16, 17, 17, 15, 16, 15, 20, 19, 19, 17, 19, 19, 20, 15, 19, 19, 16, 19, 23, 15, 18, 16, 19, 19, 14, 23, 13, 17, 21, 16, 17, 18, 18, 22, 13, 16, 20, 17, 12, 19, 18, 18, 22, 16, 16, 18, 19, 16, 17, 17, 16, 19, 17, 17, 20, 15, 19, 22, 16, 22, 15, 19, 14, 21, 18, 18, 21, 18, 18, 18, 18, 17, 16, 18, 18, 19, 19, 18, 18, 18, 17, 20, 17, 18, 17, 19, 17, 22, 19, 15, 16, 15, 18, 18, 21, 18, 21, 17, 19, 18, 16, 20, 19, 13, 14, 19, 18, 17, 20, 22, 19, 21, 18, 21, 19, 18, 13, 21, 15, 18, 18, 17, 18, 16, 15, 20, 18, 17, 13, 18, 17, 19, 19, 18, 17, 22, 18, 16, 21, 14, 18, 18, 17, 18, 23, 19, 17, 18, 19, 17, 18, 13, 17, 20, 14, 16, 17, 18, 18, 18, 16, 18, 18, 18, 21, 21, 18, 14, 20, 19, 18, 17, 17, 15, 21, 17, 17, 20, 16, 17, 16, 19, 15, 19, 20, 14, 19, 16, 18, 21, 16, 20, 16, 19, 20, 16, 19, 17, 21, 16, 21, 18, 19, 15, 21, 20, 15, 20, 19, 21, 19, 17, 15, 20, 19, 18, 16, 15, 18, 17, 18, 18, 21, 17, 18, 18, 16, 18, 22, 17, 18, 19, 18, 16, 20, 19, ...

Put as a table, the first 100 centuries are:

k	n
0	3
1	7
2	10
3	10
4	11
5	12
6	10
7	14
8	11
9	12
10	12
11	14
12	11
13	17
14	9
15	14
16	13
17	14
18	14
19	15
20	12
21	16
22	13
23	11
24	16
25	17
26	11
27	12
28	16
29	15
30	14
31	18
32	15
33	11
34	17
35	12
36	13
37	16
38	15
39	15
40	13
41	17
42	10
43	19
44	15
45	14
46	16
47	14
48	18
49	13
50	14
51	15
52	18
53	16
54	13
55	15
56	14
57	16
58	12
59	19
60	14
61	17
62	13
63	11
64	20
65	15
66	16
67	16
68	14
69	13
70	19
71	16
72	15
73	19
74	15
75	11
76	16
77	16
78	16
79	18
80	15
81	16
82	14
83	17
84	18
85	16
86	13
87	15
88	15
89	17
90	15
91	16
92	15
93	15
94	13
95	19
96	13
97	17
98	14
99	17

One can also consider the smallest centuries with n composites not divisible by 2, 3, or 5, which I will tabulate below:

n	smallest ks
0	(not possible)
1	(not possible)
2	(not possible)
3	0, ...
4	(unknown)
5	(unknown)
6	..., 394338677302163715754576644, 6228039143760643018587824345, ...
7	1, ..., 123735651314895575079, ...
8	23352869714018, 983930290209021, 2755152897458531, 6327707798392539, 8835426559173950, ..., 203860951641372730864, ...
9	14, 1908189311558, 6157376214122, 7658205745776, 8078877131667, 10137710652198, 13862924841999, 17176990713081, 18883149995801, 19441702516473, 22930638581651, 27366580054772, 28441016502165, 34640826787757, 45813173440655, 56510624356859, ..., 14688670051164208, ..., 113986721676969601, ..., 140589909841627825, ..., 156565005476073094, ...
10	2, 3, 6, 42, 194, 230, 24662691, 313114319, 776749247, 2013136112, 6569717174, 16226936180, 20473355126, 23861161886, 27524565569, 28137314864, 30609996072, 30703228932, 30730057202, 30938308631, 32111780141, 44049425619, 59164187718, 59297737052, 69898357008, ..., 1228537713709, ...
11	4, 8, 12, 23, 26, 33, 63, 75, 375, 1131, 1572, 7837, 9780, 17492, 27049, 29673, 981953, 1040840, 3145701, 6936645, 6970379, 8560196, 9016406, 9255944, 9520169, 15112118, 17187179, 27261180, 39759303, 58470222, 60393431, 69026496, 93078120, 96687828, 99524981, 101914209, 105652944, 115354109, 121978380, 124847702, 129567821, 144997771, 155326974, 170092302, ...
12	5, 9, 10, 20, 27, 35, 58, 147, 179, 810, 1158, 1416, 2033, 2232, 2297, 2660, 3054, 4508, 6635, 8237, 12303, 12463, 16166, 21728, 26640, 28029, 58514, 82325, 138302, 143576, 155249, 182004, 206582, 236156, 280235, 290367, 301551, 490343, 498444, 654080, 800232, 914537, 973409, 1080495, 1134048, 1199364, 1238217, 1256649, 1257629, 1311918, 1396323, 1599869, 1689077, 1927515, 1978332, ...
13	16, 22, 36, 40, 49, 54, 62, 69, 86, 94, 96, 119, 156, 174, 201, 215, 240, 308, 320, 323, 329, 408, 416, 424, 569, 606, 633, 636, 674, 713, 798, 806, 875, 888, 900, 923, 1098, 1122, 1130, 1247, 1317, 1670, 1799, 1833, 1871, 2303, 2676, 2847, 2891, 3122, 3177, 3213, 3442, 3543, 3922, 4065, 4203, 4676, 4784, 5249, 5415, 5595, 6441, 7393, 7488, 7883, 8021, 8412, 8444, 8603, 9290, 9843, 9903, ...
14	7, 11, 15, 17, 18, 30, 45, 47, 50, 56, 60, 68, 82, 98, 101, 122, 125, 126, 129, 131, 137, 144, 152, 153, 164, 173, 203, 207, 221, 236, 277, 285, 286, 311, 333, 335, 386, 393, 398, 419, 491, 514, 525, 536, 556, 558, 590, 596, 651, 694, 701, 709, 729, 747, 755, 796, 834, 876, 911, 926, 939, 957, 1011, 1047, 1053, 1089, 1109, 1146, 1157, 1169, 1196, 1208, 1220, 1260, 1272, 1289, 1326, 1363, 1373, 1406, 1460, 1763, 1817, 2043, 2054, 2075, 2173, 2199, 2507, 2586, 2621, 2669, 2723, 2748, 2759, 2864, ...
15	19, 29, 32, 38, 39, 44, 51, 55, 65, 72, 74, 80, 87, 88, 90, 92, 93, 107, 111, 114, 124, 134, 166, 170, 182, 183, 192, 195, 210, 211, 213, 218, 220, 227, 238, 249, 251, 261, 267, 279, 284, 291, 293, 312, 324, 342, 355, 365, 369, 387, 401, 411, 412, 427, 437, 442, 446, 458, 468, 477, 479, 488, 494, 515, 518, 531, 544, 545, 564, 581, 594, 616, 629, 671, 677, 714, 719, 722, 726, 775, 777, 785, 791, 827, 832, 861, 863, 890, 896, 945, 954, 975, 978, 984, 989, 1020, 1062, 1067, 1082, 1091, 1095, ...
16	21, 24, 28, 37, 46, 53, 57, 66, 67, 71, 76, 77, 78, 81, 85, 91, 102, 103, 104, 106, 108, 110, 113, 118, 136, 140, 145, 146, 148, 151, 157, 159, 160, 169, 177, 180, 185, 197, 198, 199, 226, 235, 241, 253, 254, 256, 257, 259, 263, 264, 266, 268, 269, 270, 272, 287, 301, 313, 315, 317, 327, 330, 336, 337, 338, 344, 345, 348, 350, 358, 359, 360, 362, 366, 367, 373, 378, 383, 389, 391, 404, 405, 420, 422, 423, 429, 440, 441, 447, 449, 453, 455, 461, 464, 471, 473, 475, 484, 498, ...
17	13, 25, 34, 41, 61, 83, 89, 97, 99, 100, 117, 120, 121, 123, 128, 130, 132, 138, 139, 143, 150, 154, 158, 161, 168, 171, 181, 184, 187, 190, 200, 209, 216, 217, 222, 225, 233, 237, 239, 246, 262, 276, 278, 280, 282, 288, 290, 294, 300, 302, 303, 305, 306, 321, 325, 326, 341, 343, 346, 347, 351, 353, 354, 363, 368, 381, 392, 395, 397, 410, 418, 426, 434, 439, 445, 451, 456, 465, 466, 476, 485, 486, 492, 497, 509, 512, 519, 522, 527, 528, 529, 530, 538, 543, 546, 549, 560, 561, 568, 571, 576, 578, 587, ...
18	31, 48, 52, 79, 84, 105, 109, 112, 116, 127, 133, 135, 141, 149, 155, 162, 175, 176, 189, 204, 205, 214, 219, 224, 228, 231, 232, 243, 244, 245, 248, 255, 258, 260, 274, 275, 292, 295, 296, 298, 309, 310, 316, 331, 334, 339, 370, 371, 374, 376, 382, 390, 396, 399, 402, 413, 415, 421, 428, 432, 435, 436, 467, 474, 481, 483, 487, 490, 495, 496, 499, 507, 511, 513, 534, 537, 539, 540, 547, 553, 557, 559, 566, 570, 575, 579, 580, 582, 583, 586, 592, 601, 603, 612, 619, 627, 638, 641, 644, 648, 653, 658, 660, 667, ...
19	43, 59, 70, 73, 95, 115, 163, 165, 167, 172, 191, 193, 202, 206, 208, 212, 223, 229, 242, 252, 281, 297, 299, 304, 318, 322, 352, 357, 364, 372, 379, 388, 400, 407, 414, 417, 425, 430, 431, 448, 452, 459, 460, 462, 463, 470, 472, 480, 482, 493, 502, 504, 508, 510, 523, 532, 533, 535, 541, 550, 551, 555, 562, 563, 585, 599, 608, 610, 617, 618, 623, 625, 630, 639, 640, 642, 647, 661, 664, 666, 670, 680, 682, 685, 688, 689, 699, 703, 706, 707, 710, 712, 715, 727, 737, 761, 766, 767, 771, 779, 780, 782, 783, 786, 787, 789, 794, 795, ...
20	64, 178, 186, 234, 247, 250, 265, 271, 289, 307, 328, 332, 361, 380, 384, 409, 443, 444, 450, 454, 478, 489, 517, 520, 526, 542, 548, 554, 611, 613, 622, 628, 631, 632, 643, 645, 649, 652, 681, 683, 696, 716, 721, 724, 725, 731, 746, 749, 751, 760, 769, 778, 784, 808, 826, 853, 873, 880, 897, 925, 940, 949, 956, 963, 966, 977, 979, 985, 1006, 1008, 1042, 1044, 1050, 1058, 1065, 1072, 1083, 1097, 1110, 1126, 1127, 1134, 1141, 1143, 1144, 1145, 1149, 1154, 1159, 1160, 1162, 1165, 1166, 1180, 1181, 1185, 1189, 1201, 1212, 1217, 1221, ...
21	142, 188, 196, 273, 283, 349, 377, 385, 394, 406, 433, 438, 457, 506, 595, 604, 605, 673, 676, 697, 732, 739, 758, 763, 800, 835, 838, 866, 868, 883, 885, 889, 910, 936, 937, 946, 961, 970, 972, 976, 981, 994, 1031, 1032, 1054, 1057, 1074, 1086, 1090, 1096, 1099, 1103, 1120, 1128, 1135, 1136, 1168, 1174, 1193, 1194, 1218, 1248, 1255, 1258, 1261, 1265, 1273, 1279, 1280, 1287, 1293, 1298, 1315, 1318, 1372, 1378, 1387, 1396, 1400, 1441, 1446, 1453, 1471, 1479, 1480, 1496, 1519, ...
22	314, 319, 340, 356, 469, 524, 574, 662, 754, 805, 814, 829, 831, 859, 881, 907, 1000, 1027, 1037, 1051, 1101, 1124, 1132, 1138, 1183, 1231, 1270, 1307, 1309, 1370, 1423, 1433, 1501, 1531, 1549, 1588, 1623, 1630, 1672, 1681, 1713, 1733, 1755, 1808, 1813, 1837, 1843, 1852, 1864, 1880, 1882, 1900, 1909, 1924, 1956, 1963, 1990, 2031, 2035, 2134, 2149, 2150, 2152, 2164, 2178, 2182, 2191, 2200, 2202, 2222, 2224, 2231, 2330, 2335, 2353, 2404, 2425, ...
23	403, 588, 598, 695, 790, 797, 916, 1060, 1075, 1204, 1240, 1390, 1430, 1651, 1675, 1751, 1809, 1869, 1885, 1904, 1913, 1942, 1987, 2042, 2067, 2123, 2127, 2278, 2290, 2311, 2317, 2322, 2338, 2449, 2464, 2471, 2505, 2530, 2562, 2590, 2644, 2698, 2743, 2758, 2773, 2779, 2862, 2869, 2874, 2887, ...
24	1801, 1831, 2374, 2503, 2545, 2611, 2656, 2659, 2665, 2956, 2989, 3020, 3079, 3262, 3394, 3445, 3481, 3574, 3598, 3607, 3658, 3811, 4024, 4051, 4381, 4390, 4453, 4484, 4597, 4617, 4630, 4806, 4819, 4849, 4852, 5023, 5072, 5317, 5374, 5386, 5404, 5423, 5434, 5533, 5551, 5614, 5616, 5632, 5890, ...
25	1559, 1621, 2734, 2833, 2935, 3679, 3703, 4133, 4276, 4534, 4582, 4771, 5314, 5347, 5443, 5464, 5956, 6100, 7087, 7090, 7177, 7495, 7546, 7621, 7930, 8056, 8224, 8293, 8455, 8995, 9103, 9136, 9667, 9739, 10114, 10279, 10339, 10402, 10606, 10759, 10828, 11194, 11400, 11461, 11524, 11539, 11656, 11773, ...
26	3709, 4921, 8728, 9796, 10720, 10852, 11953, 11992, 12907, 13495, 13885, 14401, 14497, 14968, 15304, 15427, 15997, 16132, 16564, 16718, 18361, 19513, 20308, 20848, 21235, 22543, 22807, 22903, 22945, 24037, 24844, 26134, 26197, 26378, 26404, 26824, 27025, 28405, 28408, 28462, 28522, 28567, 29455, 29533, 29575, 29788, 30478, 30571, ...
27	2683, 10048, 12727, 14443, 14680, 15619, 20452, 26257, 30232, 31072, 33439, 34345, 35455, 38086, 38794, 40441, 41194, 44119, 46627, 47305, 52111, 53986, 54121, 54430, 64495, 67015, 68083, 68797, 69721, 70060, 71260, 71602, 72418, 73051, 76213, 77026, 77713, 79462, 80014, 81118, 81661, 82492, 85477, 88069, 89695, 91015, 94348, 94678, 96898, 97324, 99679, ...
28	60850, 71299, 75652, 83674, 101527, 103438, 105916, 111772, 115558, 115816, 131194, 140872, 146713, 156418, 176101, 179332, 231769, 249472, 264763, 266338, 267481, 291379, 298348, 310513, 319273, 323791, 327508, 363397, 363562, 380893, 387175, 387799, 402937, 404548, 407878, 409723, 418159, 434083, 434356, 436135, ...

No century for n = 4 or n = 5 is yet known. For n = 7, at least one is known beyond the second century but the century begining with 12373565131489557507900 is unlikely to be the second-smallest. Dickson’s conjecture from 1904 does imply centuries with 4 or 5 composite numbers not divisible by 2, 3, or 5 exist, although none have yet been discovered.

If a century has the maximum 23 primes, there are 14 possible patterns with 5 composite numbers not divisible by 2, 3, or 5, and 6 with 3. I have not seen the figures for 22-prime centuries, but the six 23-prime patterns with 3 composite numbers not divisible by 2, 3, or 5 must allow for a century with 4 such numbers by deletion of one prime.

As a last note for this post, I will tabulate the number of centuries with n composites not divisible by 2, 3, or 5 for each hundred thousand in the first million:

Number of Centuries Containing n Composite Numbers Not Divisible by 2, 3, or 5 in Each Hundred Thousand of the First Million
n	Total # in First Million	First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
3	1	1	0	0	0	0	0	0	0	0	0
4	0	0	0	0	0	0	0	0	0	0	0
5	0	0	0	0	0	0	0	0	0	0	0
6	0	0	0	0	0	0	0	0	0	0	0
7	1	1	0	0	0	0	0	0	0	0	0
8	0	0	0	0	0	0	0	0	0	0	0
9	1	1	0	0	0	0	0	0	0	0	0
10	6	6	0	0	0	0	0	0	0	0	0
11	13	9	2	0	0	0	0	0	1	0	1
12	20	10	2	4	1	1	0	1	0	1	0
13	73	36	9	4	6	4	3	1	3	4	3
14	178	62	21	13	20	19	6	11	11	9	6
15	409	95	58	32	34	46	27	33	34	27	23
16	791	165	104	101	76	57	78	54	54	50	52
17	1,195	172	160	131	130	109	109	111	84	106	83
18	1,658	173	195	197	174	160	165	143	157	152	142
19	1,796	136	160	183	184	172	187	210	189	171	204
20	1,601	68	141	156	169	193	172	160	192	173	177
21	1,160	42	87	93	104	132	132	133	137	150	150
22	621	16	41	45	59	62	65	73	82	88	90
23	318	7	16	28	29	26	37	48	35	48	44
24	119	0	2	9	11	13	14	21	14	15	20
25	34	0	2	3	2	5	5	1	7	5	4
26	4	0	0	0	1	1	0	0	0	1	1
27	1	0	0	1	0	0	0	0	0	0	0
28	0	0	0	0	0	0	0	0	0	0	0

The pattern we see here, although much more limited than the study of primes in centuries by James Glaisher from the late 1870s, is very similar to Glaisher’s findings:

1. in the first hundred thousand centuries with seventeen and eighteen composites not divisible by 2, 3, or 5 are virtually equally the most numerous
2. in the second and third hundred thousand centuries with eighteen composites not divisible by 2, 3, or 5 are clearly the most numerous
3. in the remaining hundred thousands centuries with nineteen composites not divisible by 2, 3, or 5 are the most numerous.
• however, in the eighth and ninth hundred thousand centuries with twenty composites not divisible by 2, 3, or 5 are virtually equal in number to those with nineteen
4. in the whole first million centuries with nineteen composites not divisible by 2, 3, or 5 are most numerous, followed by those with eighteen and twenty such numbers (almost equal) and those with seventeen and twenty-one composites not divisible by 2, 3, or 5 (also almost equal).

Glaisher found that in the first million centuries with eight primes were most numerous. These may contain either eighteen or twenty composites not divisible by 2, 3, or 5. A century with nineteen composites not divisible by 2, 3, or 5 may have either seven or nine primes — the total number of seven- and nine-prime centuries in the first million is indeed greater than the total number of any other pair of prime counts separated by two. If we follow from Glaisher’s table, we would predict — though it would be extremely laborious to check with the tools I possess — that centuries with 21 composites not divisible by 2, 3, or 5 would become the most numerous from the sixth million, and centuries with 20 from the third to the fifth or sixth.

Another sign of the runaway — Melbourne’s first rainless August

Today was a rather difficult sleep for me due to the strong winds overnight. I was later to bed than I had been over the previous few days when I have been affected by a headache and went to bed extremely early a few nights ago.

Given that there is no rain in the forecast apart from today, I was eagerly awaiting rain from the moment I first briefly woke up around 7 o‘clock to go to the toilet — before getting a very interrupted sleep. I was looking for signs of rain all the time this morning and afternoon, trying half-heartedly to not discuss or look at the weather to see if it was going to rain. BOM forecasts are very unfavourable for significant rain but not so extreme as I was already anticipating from reading them about a week ago, when Melbourne was in danger of its driest July on record.

All along until around 4 P.M. I was anticipating rain, but when I finally bit the bullet and had a look at BOM’s forecast I found what I feared all along — that Melbourne was not going to get rainfall predicted with a ninety percent chance in the early morning forecast. There is no confident rain in the weekly forecast either, and I have no desire checking further ahead as the seasonal outlook is for very dry and hot weather.

Probable Weather for Remainder of 2023:

1. Melbourne will have its first ever rainless August
• the previous record dry August is 12.4 millimetres in 1903
• in fact there has never previously been an August with no rain in the first eight days, but none is forecasted from Saturday until well after that
2. Melbourne will not get any rainfall in September or October either, or maximally a small fraction of the sum of the monthly record lows of 19.4 millimetres
3. Melbourne’s dry spell record of forty days — ironically set following a record wet spell of 374 millimetres in 42 days and during the city’s wettest ever twelve months with 1,045.5 millimetres between October 1954 and September 1955 — will be beaten by a large margin during the normally rainy winter months
• if Melbourne beats it by the margin Sydney did in 1995 — 47 days vis-à-vis a previous record of 34 — Melbourne would go without rain for at least 55 days from 1 August until 24 September
• Even during the extreme droughts of 1914, 1982 and 2006, there was no rainless spell longer than 28 days
4. Melbourne’s annual rainfall will be below 332 millimetres virtually every year from 2024, and below 166 millimetres most years
5. Melbourne’s dams will be permanently dry soon after 2024, with the last runoff-producing rain having occurred this June

It has long occurred to me that runaway poleward movement of the frontal systems to well south of Tasmania is long overdue based on circulation changes observed since 1964. These changes would, as I noted four-and-a-half years ago, mean that Melbourne receives no more than 180 millimetres of rainfall per year — and possibly given topographic effects much less than that. With more frequent positive Indian Ocean Dipoles, that figure should decline still further to under 100 millimetres per year.

Such a rapid “catch-up decline” would put an end to most of the attractive feature of Melbourne that the Public Transport Users’ Association noted as already threatened a quarter of a century ago. It shows that runaway climate change was already a danger then. Moreover, the PTUA’s proposals were in fact not nearly radical enough. What was needed was 100 percent reduction in car use and a 100 percent reduction in Australia emissions no later than 2005. This was and is — via rational planning and expropriation of the rich polluters — achievable without mobility loss and indeed with mobility gain by eliminating congestion. In addition, even if this rational planning cannot does not reverse runaway climate change and the transformation of Melbourne into a hyperarid desert, it will mean that the polluters responsible are made to pay the cost.

The thing that must never be said – it overlooks the issue

Today, Mummy and I went on a train ride to pick up the new Peugeot, which had a major fault in its brake control that had to be repaired at a workshop near the bend of Hawthorn Road into Camberwell Road. In earlier years, this area was familiar, but since COVID we have never been there.

On the whole, I enjoyed the return trip with my mother to pick up the car — a 2018 Peugeot 208 which was bought in Balwyn last summer. I was really very reactive to any complaint about the standard of Melbourne’s public transport, which by global standards is exceedingly bad. There remains the problem of loud, one-way “mantras” about the need to end all spending on roads and to transfer it to public transport. Nevertheless, so early as 1990, Environment, Capitalism and Socialism demonstrated the potential benefits to the immense majority of such a policy, while much more recently Richard Smith had demonstrated the urgent necessity of radically reducing car production and banning the manufacture of all but the smallest road vehicles.

Given what Smith reveals about the energy costs of car production, one must never say that cars are a necessary benefit when criticising Australia’s woeful public transport. Cars’ environmental cost is far too great to, even given the standard of public transport in Australia, excuse their use.

The fact is — as known for over three decades — that a rational plan could with 100 percent public transport modal share and resultant vastly reduced greenhouse emissions and pollution nevertheless produce the same (or greater) mobility cars provide, and at far less financial cost than the sum currently wasted on roads. Plainly put, such a plan is in every way except reduction in profit to wealthy corporations much fairer, cheaper and more rational than the present transport system. There should be no tolerance (personally or politically) for any policy other than 100 percent of spending on public transport, and not a solitary cent on roads.

Consistent failure and consistent silence

Probably the most striking and most consistent feature of cricket history — at least since the middle 1890s — has been the failure of English spin bowlers in Australia. This is especially true given their dominance of first-class bowling records.

Since Bobby Peel and Johnny Briggs in the late 1880s and early 1890s — as will be noted, a wholly unsuitable comparison — no English spin bowler has ever been a matchwinner at Test level in Australia. The universal failure of the most successful county bowlers like Tich Freeman, Wilfrid Rhodes, Colin Blythe, Eric Hollies, Frank Woolley or Richard Tyldesley, plus the fact that an even greater number of extremely successful county bowlers like Charlie Parker, Tom Goddard, George Dennett, Walter Mead, Schofield Haigh, Albert Hallam and Razor Smith were never remotely considered for a tour of Australia, is chilling. Even the less unsuccessful English spinners — Roy Kilner, Jack White, Hedley Verity, Jim Laker, Fred Titmus, Derek Underwood — were never able to win Tests like they consistently won county matches at home. What these relative successes could do was contain batsmen over extremely long spells, a critical skill in the conditions of Test cricket in Australia at the time.

Statistics demonstrate the abysmal record of English spinners in Australia:

1. since 1905, only twice has an English spinner taken ten wickets in a Test in Australia
1. out of more than forty occurrences in all Ashes Tests
2. and many more than that elsewhere in international cricket
2. no English spinner had taken so many as 26 wickets in a series in Australia since 1903/1904
1. whereas as of 1998 (a quarter of a century ago!) this had occurred more than seventy times in all Test cricket

Both 1903/1904, when Rhodes took thirty-one wickets in five Tests, and the success of Peel and Briggs between 1886 and 1895, reflect abnormal climatic conditions. 1886 to 1894 was the wettest era in the instrumental record in eastern Australia, and in Melbourne 1903/1904 was the wettest and second-coolest summer since records began in 1856. The wet conditions produced pitches much less unfavourable to spin.

What is equally striking, analysing both Wisden and the English and Australian presses, is the complete absence until the 1950s of discussion about English spinners’ abysmal record in Australia. Whenever discussing whom to choose for an Australian tour the failure of English spin was not discussed. Nor was the question ever asked after the most unsuccessful Ashes tours like 1907/1908, 1920/1921, 1924/1925 and 1946/1947. The Australian press was almost equally silent, although it might be expected to have more idea why English spinners were so hopeless. The fact that Australia produced an exceptional number of high-class spinners in this very period constitutes the proof Australian pitches were not impossible for spin bowling.

Two obvious reasons are:

1. the sport-watching public possessed negligible interest in understanding why English spin bowling so consistently failed in Australia
2. because of the large number of first-class teams in England, bowlers much better suited to Australian pitches always existed, making the universal failure of English spin into a non-issue

However, I have recently thought that a more important reason for the silence at least of Wisden and the English press may be that at a very early point — following Colin Blythe’s complete failure in 1907/1908 — it became accepted that failure of English spinners in Australia was a law of nature and that expecting them to win matches as they did in England was tantamount to tampering with nature. Belief that either:

1. English spinners could never be trained or coached to emulate their English successes in Australia because their special skills were specific to English conditions and entirely useless in Australia, or
2. the cost of making English spinners as effective in Australia as at home would be prohibitive in one or more of:
1. labour
2. time
3. their value at home via either
• reduced success in English conditions
• loss of the art of spin via adopting medium pace in Australia, as Bobby Peel and Freddie Brown did

fits perfectly with the silence about the abysmal record of English spin in Australia. Belief that English spinners could never be effective in Australia (under normal climatic conditions) would explain why the 1911/1912 and 1920/1921 teams included no player for spin alone, and the surprise of Wisden at the objectively modest but comparatively exceptional success of Jack White in 1928/1929.

When England’s writers became nostalgic for predominant spin bowling in the 1960s, foreign writers occasionally discussed English spinners’ woeful record in Australia, if without the vehemence with which Wisden writers criticised current English cricket, and rarely quoting the striking figures noted at the beginning of this post. It is likely that writers in Australia and the West Indies viewed even the very best English spinners as mediocre-to-bad bowlers unimportant to the broader history of cricket. (This, even if accurate, meant they failed to dissect their weaknesses). English writers, for their part, were tied between their nostalgia for prominent spin bowling and the plain fact that English spinners had been matchwinners in Test cricket only at home, and there only before covering was adopted in 1959. In both cases the presumed perspective was highly unfavourable to dissecting and analysing why England could not produce spin bowlers capable of sustained success in Tests under Australian conditions.

Four sub-sequences revealed

Notice the following four complete sequences, in which the next number satisfying n after zero is listed if zero satisfies the requirement:

0	1	2	3	4	5		6		7
104	29	3	5	9	0	1	42		80562
132	31	9	4	3	1		2054		0	190783
89	21	10	2	5	22		0	1	224241
135	24	6	2	3	0	1	4		211696

These four sequences are, in a sense, sub-sequences of A186311. Numbers being each row from the top, we have:

1. Least k such that the interval 100k to 100k+99 has exactly n primes ending in 1
2. Least k such that the interval 100k to 100k+99 has exactly n primes ending in 3
3. Least k such that the interval 100k to 100k+99 has exactly n primes ending in 7
4. Least k such that the interval 100k to 100k+99 has exactly n primes ending in 9

It is easy to see that there can never be more than seven primes ending in the same digit between multiples of 100, as minimally three numbers must divide by 3. When I looked more closely it seemed clear that there could be six primes in a century ending in the same digit, but I was unsure that there could be seven. Brian Kehrig, in a LinkedIn conservation, said there could be and gave the smallest example after the first century (all factorisations courtesy of Alpertron, although I have expanded repeated factors rather than use exponents):

1. 8056201 is prime
8056203 = 3 × 2 685401
8056207 = 73 × 110359
8056209 = 3 × 7 × 19 × 61 × 331
2. 8056211 is prime
8056213 = 11 × 41 × 17863
8056217 = 13 × 467 × 1327
8056219 is prime
8056221 = 3 × 2685407
8056223 = 7 × 47 × 47 × 521
8056227 = 3 × 2685409
8056229 = 29 × 83 × 3347
3. 8056231 is prime
8056233 = 3 × 3 × 3 × 23 × 12973
8056237 = 7 × 7 × 164413
8056239 = 3 × 571 × 4703
4. 8056241 is prime
8056243 = 13 × 619711
8056247 = 19 × 167 × 2539
8056249 = 17 × 31 × 15287
8056251 = 3 × 3 × 7 × 127877
8056253 = 179 × 45007
8056257 = 3 × 11 × 244129
8056259 is prime
5. 8056261 is prime
8056263 = 3 × 1609 × 1669
8056267 is prime
8056269 = 3 × 3 × 13 × 37 × 1861
6. 8056271 is prime
8056273 = 59 × 136547
8056277 is prime
8056279 = 7 × 11 × 23 × 4549
8056281 = 3 × 67 × 149 × 269
8056283 = 17 × 473899
8056287 = 3 × 3 × 3 × 29 × 10289
8056289 is prime
7. 8056291 is prime
8056293 = 3 × 7 × 383633
8056297 is prime

It’s notable that a(7) for all these sequences lies in the “core” of the “eight-digit gap” where the geometric distance between prime-rich centuries is at a maximum. Within the numbers encompassing these centuries there is no century containing so many as fifteen total primes. The 80,563^rd century is the 263^rd with thirteen total primes, while the 190,784^th, 211,697^th, and 224,242^nd all have too few primes to be similarly counted on OEIS.

The possibility of these sequences I observed many years ago when I noticed that all the following numbers were composite:

• 8907 = 3 × 2969
• 8917 = 37 × 241
• 8927 = 79 × 113
• 8937 = 3 × 3 × 3 × 331
• 8947 = 23 × 389
• 8957 = 13 × 13 × 53
• 8967 = 3 × 7 × 7 × 61
• 8977 = 47 × 191
• 8987 = 11 × 19 × 43
• 8997 = 3 × 2999

I also noticed at a fairly early stage the unusual factorisations of the 105^th century:

10401 = 3 × 3467
10403 = 101 × 103
10407 = 3 × 3469
10409 = 7 × 1487
10411 = 29 × 359
10413 = 3 × 3 × 13 × 89
10417 = 11 × 947
10419 = 3 × 23 × 151
10421 = 17 × 613
10423 = 7 × 1489
1. 10427 is prime
2. 10429 is prime
10431 = 3 × 3 × 19 × 61
3. 10433 is prime
10437 = 3 × 7 × 7 × 71
10439 = 11 × 13 × 73
10441 = 53 × 197
10443 = 3 × 59 × 59
10447 = 31 × 337
10449 = 3 × 3 × 3 × 3 × 3 × 43
10451 = 7 × 1493
4. 10453 is prime
5. 10457 is prime
6. 10459 is prime
10461 = 3 × 11 × 317
7. 10463 is prime
10467 = 3 × 3 × 1163
10469 = 19 × 19 × 29
10471 = 37 × 283
10473 = 3 × 3491
8. 10477 is prime
10479 = 3 × 7 × 499
10481 = 47 × 223
10483 = 11 × 953
9. 10487 is prime
10489 = 17 × 617
10491 = 3 × 13 × 269
10493 = 7 × 1499
10497 = 3 × 3499
10. 10499 is prime

The four A186311 “sub-sequences” are not yet on OEIS, nor is there any OEIS list of centuries with seven primes ending in one of 1, 3, 7, or 9. Given that I would need better software and an OEIS account to add them, the brief discussion here is something I have wanted to do.