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 |
| n | smallest ks |
|---|---|
| 0 | (not possible) |
| 1 | (not possible) |
| 2 | (not possible) |
| 3 | 0, ... |
| 4 | (unknown) |
| 5 | (unknown) |
| 6 | ..., 394338677302163715754576644, 6228039143760643018587824345, ... |
| 7 | 1, ... |
| 8 | 23352869714018, 983930290209021, ... |
| 9 | 14, 1908189311558, 6157376214122, 7658205745776, 8078877131667, 10137710652198, 13862924841999, 17176990713081, 18883149995801, 19441702516473, 22930638581651, 27366580054772, 28441016502165, 34640826787757, 45813173440655, 56510624356859, ..., 14688670051164208, ... |
| 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, ... |
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 |
- in the first hundred thousand centuries with seventeen and eighteen composites not divisible by 2, 3, or 5 are virtually equally the most numerous
- in the second and third hundred thousand centuries with eighteen composites not divisible by 2, 3, or 5 are clearly the most numerous
- 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
- 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.
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