Illustrating each count of composites not divisible by 2, 3, or 5

Following on from my previous post, I will now illustrate each count of composite numbers no divisible by 2, 3, or 5 in a century, except for those not known to occur.

Some of these tables have been shown on this blog at ‘The 16,719th century and other prime-free sequences’ from almost a decade ago.

For a few counts affected by the “eight-digit gap” — those from nine to eleven — I will show those centuries immediately below and above the “eight-digit gap” in a subsequent table for more perspective. Centuries with fewer than nine composites not divisible by 2, 3, or 5 also show the effect of the “eight-digit gap”, but this is obscured because no century below the eight-digit gap has six or eight composites not divisible by 2, 3, or 5, whilst none with seven larger than the second century are known — and of course none with the theoretically possible four or five composites not divisible by 2, 3, or 5 have yet been found.

n	Factorisations of numbers not divisible by 2, 3, or 5
3	7 is prime 11 is prime 13 is prime 17 is prime 19 is prime 23 is prime 29 is prime 31 is prime 37 is prime 41 is prime 43 is prime 47 is prime 1. 49 = 7 × 7 53 is prime 59 is prime 61 is prime 67 is prime 71 is prime 73 is prime 2. 77 = 7 × 11 79 is prime 83 is prime 89 is prime 3. 91 = 7 × 13 97 is prime
4	(none known)
5	(none known)
6	39433867730216371575457664401 is prime 39433867730216371575457664407 is prime 39433867730216371575457664411 is prime 39433867730216371575457664413 is prime 39433867730216371575457664417 is prime 1. 39433867730216371575457664419 = 7 × 19 × 23 × 229 × 2179 × 25834328988233247551 39433867730216371575457664423 is prime 39433867730216371575457664429 is prime 39433867730216371575457664431 is prime 39433867730216371575457664437 is prime 39433867730216371575457664441 is prime 39433867730216371575457664443 is prime 2. 39433867730216371575457664447 = 7 × 11 × 29 × 993961 × 17766885682653834719 39433867730216371575457664449 is prime 39433867730216371575457664453 is prime 39433867730216371575457664459 is prime 3. 39433867730216371575457664461 = 7 × 13 × 233 × 6050207 × 307398559703912641 39433867730216371575457664467 is prime 39433867730216371575457664471 is prime 39433867730216371575457664473 is prime 39433867730216371575457664477 is prime 39433867730216371575457664479 is prime 39433867730216371575457664483 is prime 4. 39433867730216371575457664489 = 7 × 17 × 9470228338819 × 34991451936949 5. 39433867730216371575457664491 = 11 × 47 × 432569 × 331736687 × 531532637641 6. 39433867730216371575457664497 = 389 × 547968840241 × 184996669834253
7	101 is prime 103 is prime 107 is prime 109 is prime 113 is prime 1. 119 = 7 × 17 2. 121 = 11 × 11 127 is prime 131 is prime 3. 133 = 7 × 19 137 is prime 139 is prime 4. 143 = 11 × 13 149 is prime 151 is prime 157 is prime 5. 161 = 7 × 23 163 is prime 167 is prime 6. 169 = 13 × 13 173 is prime 179 is prime 181 is prime 7. 187 = 11 × 17 191 is prime 193 is prime 197 is prime 199 is prime
8	2335286971401803 is prime 2335286971401809 is prime 1. 2335286971401811 = 7 × 137 × 9433 × 258149813 2. 2335286971401817 = 37 × 89 × 257 × 1319 × 2092043 2335286971401821 is prime 2335286971401823 is prime 2335286971401827 is prime 2335286971401829 is prime 3. 2335286971401833 = 83 × 83 × 409 × 2153 × 384961 4. 2335286971401839 = 7 × 7 × 13 × 17 × 29 × 41 × 47 × 313 × 12329 2335286971401841 is prime 2335286971401847 is prime 2335286971401851 is prime 5. 2335286971401853 = 7 × 79 × 5413 × 26711 × 29207 6. 2335286971401857 = 11 × 127 × 1671644217181 2335286971401859 is prime 2335286971401863 is prime 2335286971401869 is prime 2335286971401871 is prime 2335286971401877 is prime 7. 2335286971401881 = 7 × 19 × 125507 × 139900951  2335286971401883 is prime 2335286971401887 is prime 2335286971401889 is prime 8. 2335286971401893 = 35255371 × 66239183  2335286971401899 is prime
9	1. 1403 = 23 × 61 1409 is prime 2. 1411 = 17 × 83 3. 1417 = 13 × 109 4. 1421 = 7 × 7 × 29 1423 is prime 1427 is prime 1429 is prime 1433 is prime 1439 is prime 5. 1441 = 11 × 131 1447 is prime 1451 is prime 1453 is prime 6. 1457 = 31 × 47 1459 is prime 7. 1463 = 7 × 11 × 19 8. 1469 = 13 × 113 1471 is prime 9. 1477 = 7 × 211 1481 is prime 1483 is prime 1487 is prime 1489 is prime 1493 is prime 1499 is prime
10	1. 203 = 7 × 29 2. 209 = 11 × 19 211 is prime 3. 217 = 7 × 31 4. 221 = 13 × 17 223 is prime 227 is prime 229 is prime 233 is prime 239 is prime 241 is prime 5. 247 = 13 × 19 251 is prime 6. 253 = 11 × 23 257 is prime 7. 259 = 7 × 37 263 is prime 269 is prime 271 is prime 277 is prime 281 is prime 283 is prime 8. 287 = 7 × 41 9. 289 = 17 × 17 293 is prime 10. 299 = 13 × 23
11	401 is prime 1. 403 = 13 × 31 2. 407 = 11 × 37 409 is prime 3. 413 = 7 × 59 419 is prime 421 is prime 4. 427 = 7 × 61 431 is prime 433 is prime 5. 437 = 19 × 23 439 is prime 443 is prime 449 is prime 6. 451 = 11 × 41 457 is prime 461 is prime 463 is prime 467 is prime 7. 469 = 7 × 67 8. 473 = 11 × 43 479 is prime 9. 481 = 13 × 37 487 is prime 491 is prime 10. 493 = 17 × 29 11. 497 = 7 × 71 499 is prime
12	503 is prime 509 is prime 1. 511 = 7 × 73 2. 517 = 11 × 47 521 is prime 523 is prime 3. 527 = 17 × 31 4. 529 = 23 × 23 5. 533 = 13 × 41 6. 539 = 7 × 7 × 11 541 is prime 547 is prime 7. 551 = 19 × 29 8. 553 = 7 × 79 557 is prime 9. 559 = 13 × 43 563 is prime 569 is prime 571 is prime 577 is prime 10. 581 = 7 × 83 11. 583 = 11 × 53 587 is prime 12. 589 = 19 × 31 593 is prime 599 is prime
13	1601 is prime 1. 1603 = 7 × 229 1607 is prime 1609 is prime 1613 is prime 1619 is prime 1621 is prime 1627 is prime 2. 1631 = 7 × 233 3. 1633 = 23 × 71 1637 is prime 4. 1639 = 11 × 149 5. 1643 = 31 × 53 6. 1649 = 17 × 97 7. 1651 = 13 × 127 1657 is prime 8. 1661 = 11 × 151 1663 is prime 1667 is prime 1669 is prime 9. 1673 = 7 × 239 10. 1679 = 23 × 73 11. 1681 = 41 × 41 12. 1687 = 7 × 241 13. 1691 = 19 × 89 1693 is prime 1697 is prime 1699 is prime
14	701 is prime 1. 703 = 19 × 37 2. 707 = 7 × 101 709 is prime 3. 713 = 23 × 31 719 is prime 4. 721 = 7 × 103 727 is prime 5. 731 = 17 × 43 733 is prime 6. 737 = 11 × 67 739 is prime 743 is prime 7. 749 = 7 × 107 751 is prime 757 is prime 761 is prime 8. 763 = 7 × 109 9. 767 = 13 × 59 769 is prime 773 is prime 10. 779 = 19 × 41 11. 781 = 11 × 71 787 is prime 12. 791 = 7 × 113 13. 793 = 13 × 61 797 is prime 14. 799 = 17 × 47
15	1901 is prime 1. 1903 = 11 × 173 1907 is prime 2. 1909 = 23 × 83 1913 is prime 3. 1919 = 19 × 101 4. 1921 = 17 × 113 5. 1927 = 41 × 47 1931 is prime 1933 is prime 6. 1937 = 13 × 149 7. 1939 = 7 × 277 8. 1943 = 29 × 67 1949 is prime 1951 is prime 9. 1957 = 19 × 103 10. 1961 = 37 × 53 11. 1963 = 13 × 151 12. 1967 = 7 × 281 13. 1969 = 11 × 179 1973 is prime 1979 is prime 14. 1981 = 7 × 283 1987 is prime 15. 1991 = 11 × 181 1993 is prime 1997 is prime 1999 is prime
16	1. 2101 = 11 × 191 2. 2107 = 7 × 7 × 43 2111 is prime 2113 is prime 3. 2117 = 29 × 73 4. 2119 = 13 × 163 5. 2123 = 11 × 193 2129 is prime 2131 is prime 2137 is prime 2141 is prime 2143 is prime 6. 2147 = 19 × 113 7. 2149 = 7 × 307 2153 is prime 8. 2159 = 17 × 127 2161 is prime 9. 2167 = 11 × 197 10. 2171 = 13 × 167 11. 2173 = 41 × 53 12. 2177 = 7 × 311 2179 is prime 13. 2183 = 37 × 59 14. 2189 = 11 × 199 15. 2191 = 7 × 313 16. 2197 = 13 × 13 × 13
17	1301 is prime 1303 is prime 1307 is prime 1. 1309 = 7 × 11 × 17 2. 1313 = 13 × 101 1319 is prime 1321 is prime 1327 is prime 3. 1331 = 11 × 11 × 11 4. 1333 = 31 × 43 5. 1337 = 7 × 191 6. 1339 = 13 × 103 7. 1343 = 17 × 79 8. 1349 = 19 × 71 9. 1351 = 7 × 193 10. 1357 = 23 × 59 1361 is prime 11. 1363 = 29 × 47 1367 is prime 12. 1369 = 37 × 37 1373 is prime 13. 1379 = 7 × 197 1381 is prime 14. 1387 = 19 × 73 15. 1391 = 13 × 107 16. 1393 = 7 × 199 17. 1397 = 11 × 127 1399 is prime
18	1. 3101 = 7 × 443 2. 3103 = 29 × 107 3. 3107 = 13 × 239 3109 is prime 4. 3113 = 11 × 283 3119 is prime 3121 is prime 5. 3127 = 53 × 59 6. 3131 = 31 × 101 7. 3133 = 13 × 241 3137 is prime 8. 3139 = 43 × 73 9. 3143 = 7 × 449 10. 3149 = 47 × 67 11. 3151 = 23 × 137 12. 3157 = 7 × 11 × 41 13. 3161 = 29 × 109 3163 is prime 3167 is prime 3169 is prime 14. 3173 = 19 × 167 15. 3179 = 11 × 17 × 17 3181 is prime 3187 is prime 3191 is prime 16. 3193 = 31 × 103 17. 3197 = 23 × 139 18. 3199 = 7 × 457
19	1. 4301 = 11 × 17 × 23 2. 4303 = 13 × 331 3. 4307 = 59 × 73 4. 4309 = 31 × 139 5. 4313 = 19 × 227 6. 4319 = 7 × 617 7. 4321 = 29 × 149 4327 is prime 8. 4331 = 61 × 71 9. 4333 = 7 × 619 4337 is prime 4339 is prime 10. 4343 = 43 × 101 4349 is prime 11. 4351 = 19 × 229 4357 is prime 12. 4361 = 7 × 7 × 89 4363 is prime 13. 4367 = 11 × 397 14. 4369 = 17 × 257 4373 is prime 15. 4379 = 29 × 151 16. 4381 = 13 × 337 17. 4387 = 41 × 107 4391 is prime 18. 4393 = 23 × 191 4397 is prime 19. 4399 = 53 × 83
20	1. 6401 = 37 × 173 2. 6403 = 19 × 337 3. 6407 = 43 × 149 4. 6409 = 13 × 17 × 29 5. 6413 = 11 × 11 × 53 6. 6419 = 7 × 7 × 131 6421 is prime 6427 is prime 7. 6431 = 59 × 109 8. 6433 = 7 × 919 9. 6437 = 41 × 157 10. 6439 = 47 × 137 11. 6443 = 17 × 379 6449 is prime 6451 is prime 12. 6457 = 11 × 587 13. 6461 = 7 × 13 × 71 14. 6463 = 23 × 281 15. 6467 = 29 × 223 6469 is prime 6473 is prime 16. 6479 = 11 × 19 × 31 6481 is prime 17. 6487 = 13 × 499 6491 is prime 18. 6493 = 43 × 151 19. 6497 = 73 × 89 20. 6499 = 67 × 97
21	1. 14201 = 11 × 1291 2. 14203 = 7 × 2029 14207 is prime 3. 14209 = 13 × 1093 4. 14213 = 61 × 233 5. 14219 = 59 × 241 14221 is prime 6. 14227 = 41 × 347 7. 14231 = 7 × 19 × 107 8. 14233 = 43 × 331 9. 14237 = 23 × 619 10. 14239 = 29 × 491 14243 is prime 14249 is prime 14251 is prime 11. 14257 = 53 × 269 12. 14261 = 13 × 1097 13. 14263 = 17 × 839 14. 14267 = 11 × 1297 15. 14269 = 19 × 751 16. 14273 = 7 × 2039 17. 14279 = 109 × 131 14281 is prime 18. 14287 = 7 × 13 × 157 19. 14291 = 31 × 461 14293 is prime 20. 14297 = 17 × 29 × 29 21. 14299 = 79 × 181
22	1. 31403 = 31 × 1013 2. 31409 = 7 × 7 × 641 3. 31411 = 101 × 311 4. 31417 = 89 × 353 5. 31421 = 13 × 2417 6. 31423 = 7 × 67 × 67 7. 31427 = 11 × 2857 8. 31429 = 53 × 593 9. 31433 = 17 × 43 × 43 10. 31439 = 149 × 211 11. 31441 = 23 × 1367 12. 31447 = 13 × 41 × 59 13. 31451 = 7 × 4493 14. 31453 = 71 × 443 15. 31457 = 83 × 379 16. 31459 = 163 × 193 17. 31463 = 73 × 431 31469 is prime 18. 31471 = 11 × 2861 31477 is prime 31481 is prime 19. 31483 = 19 × 1657 20. 31487 = 23 × 37 × 37 31489 is prime 21. 31493 = 7 × 11 × 409 22. 31499 = 13 × 2423
23	1. 40301 = 191 × 211 2. 40303 = 41 × 983 3. 40307 = 17 × 2371 4. 40309 = 173 × 233 5. 40313 = 7 × 13 × 443 6. 40319 = 23 × 1753 7. 40321 = 61 × 661 8. 40327 = 7 × 7 × 823 9. 40331 = 31 × 1301 10. 40333 = 53 × 761 11. 40337 = 11 × 19 × 193 12. 40339 = 13 × 29 × 107 40343 is prime 13. 40349 = 157 × 257 40351 is prime 40357 is prime 40361 is prime 14. 40363 = 181 × 223 15. 40367 = 37 × 1091 16. 40369 = 7 × 73 × 79 17. 40373 = 47 × 859 18. 40379 = 149 × 271 19. 40381 = 11 × 3671 40387 is prime 20. 40391 = 13 × 13 × 239 21. 40393 = 31 × 1303 22. 40397 = 7 × 29 × 199 23. 40399 = 71 × 569
24	1. 180101 = 19 × 9479 2. 180103 = 7 × 11 × 2339 3. 180107 = 389 × 463 4. 180109 = 233 × 773 5. 180113 = 23 × 41 × 191 6. 180119 = 29 × 6211 7. 180121 = 281 × 641 8. 180127 = 43 × 59 × 71 9. 180131 = 7 × 25733 10. 180133 = 61 × 2953 180137 is prime 11. 180139 = 19 × 19 × 499 12. 180143 = 151 × 1193 13. 180149 = 17 × 10597 14. 180151 = 47 × 3833 15. 180157 = 257 × 701 180161 is prime 16. 180163 = 67 × 2689 17. 180167 = 13 × 13859 18. 180169 = 11 × 11 × 1489 19. 180173 = 7 × 7 × 3677 180179 is prime 180181 is prime 20. 180187 = 7 × 25741 21. 180191 = 11 × 16381 22. 180193 = 13 × 83 × 167 23. 180197 = 367 × 491 24. 180199 = 79 × 2281
25	1. 155903 = 11 × 14173 2. 155909 = 13 × 67 × 179 3. 155911 = 7 × 22273 4. 155917 = 23 × 6779 155921 is prime 5. 155923 = 41 × 3803 6. 155927 = 241 × 647 7. 155929 = 211 × 739 8. 155933 = 19 × 29 × 283 9. 155939 = 7 × 22277 10 .155941 = 17 × 9173 11. 155947 = 11 × 14177 12. 155951 = 277 × 563 13. 155953 = 7 × 22279 14. 155957 = 83 × 1879 15. 155959 = 263 × 593 16. 155963 = 23 × 6781 17. 155969 = 11 × 11 × 1289 18. 155971 = 19 × 8209 19. 155977 = 61 × 2557 20. 155981 = 7 × 22283 21. 155983 = 151 × 1033 22. 155987 = 13 × 13 × 13 × 71 23. 155989 = 389 × 401 24. 155993 = 47 × 3319 25. 155999 = 257 × 607
26	1. 370901 = 421 × 881 2. 370903 = 13 × 103 × 277 3. 370907 = 167 × 2221 4. 370909 = 7 × 11 × 4817 5. 370913 = 73 × 5081 370919 is prime 6. 370921 = 23 × 16127 7. 370927 = 41 × 83 × 109 8. 370931 = 11 × 33721 9. 370933 = 59 × 6287 10. 370937 = 7 × 19 × 2789 11. 370939 = 29 × 12791 12. 370943 = 347 × 1069 370949 is prime 13. 370951 = 7 × 197 × 269 14. 370957 = 17 × 21821 15. 370961 = 43 × 8627 16. 370963 = 409 × 907 17. 370967 = 23 × 127 × 127 18. 370969 = 107 × 3467 19. 370973 = 101 × 3673 20. 370979 = 7 × 7 × 67 × 113 21. 370981 = 13 × 28537 22. 370987 = 349 × 1063 23. 370991 = 17 × 139 × 157 24. 370993 = 7 × 52999 25. 370997 = 11 × 29 × 1163 26. 370999 = 37 × 37 × 271
27	1. 268301 = 11 × 24391 2. 268303 = 7 × 38329 3. 268307 = 13 × 20639 4. 268309 = 71 × 3779 5. 268313 = 157 × 1709 6. 268319 = 251 × 1069 7. 268321 = 179 × 1499 8. 268327 = 151 × 1777 9. 268331 = 7 × 38333 10. 268333 = 13 × 20641 11. 268337 = 19 × 29 × 487 12. 268339 = 53 × 61 × 83 268343 is prime 13. 268349 = 149 × 1801 14. 268351 = 127 × 2113 15. 268357 = 101 × 2657 16. 268361 = 37 × 7253 17. 268363 = 43 × 79 × 79 18. 268367 = 11 × 31 × 787 19. 268369 = 167 × 1607 20. 268373 = 7 × 7 × 5477 21. 268379 = 17 × 15787 22. 268381 = 349 × 769 23. 268387 = 7 × 23 × 1667 24. 268391 = 59 × 4549 25. 268393 = 311 × 863 26. 268397 = 239 × 1123 27. 268399 = 97 × 2767
28	1. 6085001 = 13 × 223 × 2099 2. 6085003 = 139 × 43777 3. 6085007 = 67 × 90821 4. 6085009 = 7 × 241 × 3607 5. 6085013 = 11 × 641 × 863 6. 6085019 = 89 × 68371 7. 6085021 = 31 × 196291 8. 6085027 = 13 × 468079 9. 6085031 = 17 × 263 × 1361 10. 6085033 = 313 × 19441 11. 6085037 = 7 × 869291 12. 6085039 = 181 × 33619 13. 6085043 = 47 × 129469 14. 6085049 = 269 × 22621 15. 6085051 = 7 × 869293 16. 6085057 = 11 × 37 × 14951 17. 6085061 = 73 × 83357 18. 6085063 = 907 × 6709 19. 6085067 = 881 × 6907 20. 6085069 = 191 × 31859 21. 6085073 = 19 × 320267 22. 6085079 = 7 × 11 × 13 × 6079 23. 6085081 = 131 × 46451 24. 6085087 = 23 × 23 × 11503 25. 6085091 = 1013 × 6007 26. 6085093 = 7 × 869299 27. 6085097 = 41 × 193 × 769 28. 6085099 = 17 × 29 × 12343

Surrounding Centuries Not Shown for Composite Counts Affected by Eight-Digit Gap

n	Factorisations of numbers not divisible by 2, 3, or 5
9	190818931155803 is prime 1. 190818931155809 = 11 × 43 × 54851 × 7354883 2. 190818931155811 = 7 × 7 × 3894263901139 190818931155817 is prime 190818931155821 is prime 190818931155823 is prime 3. 190818931155827 = 13 × 1259 × 11658760381 4. 190818931155829 = 37 × 983 × 5246458199 190818931155833 is prime 5. 190818931155839 = 7 × 17 × 19 × 887 × 95147477 190818931155841 is prime 6. 190818931155847 = 59 × 73 × 9883 × 4482887 190818931155851 is prime 7. 190818931155853 = 7 × 11 × 13 × 499 × 382020647 190818931155857 is prime 190818931155859 is prime 190818931155863 is prime 190818931155869 is prime 190818931155871 is prime 8. 190818931155877 = 19 × 1571 × 56533 × 113081 9. 190818931155881 = 7 × 23 × 84509 × 14024669 190818931155883 is prime 190818931155887 is prime 190818931155889 is prime 190818931155893 is prime 190818931155899 is prime
10	23003 is prime 1. 23009 = 7 × 19 × 173 23011 is prime 23017 is prime 23021 is prime 2. 23023 = 7 × 11 × 13 × 23 23027 is prime 23029 is prime 3. 23033 = 31 × 743 23039 is prime 23041 is prime 4. 23047 = 19 × 1213 5. 23051 = 7 × 37 × 89 23053 is prime 23057 is prime 23059 is prime 23063 is prime 6. 23069 = 17 × 23 × 59 23071 is prime 7. 23077 = 47 × 491 23081 is prime 8. 23083 = 41 × 563 23087 is prime 9. 23089 = 11 × 2099 10. 23093 = 7 × 3299 23099 is prime
	1. 2466269101 = 17 × 61 × 2378273 2466269107 is prime 2466269111 is prime 2. 2466269113 = 7 × 11 × 1201 × 26669 3. 2466269117 = 13 × 47 × 971 × 4157 2466269119 is prime 2466269123 is prime 4. 2466269129 = 67 × 2269 × 16223 2466269131 is prime 2466269137 is prime 5. 2466269141 = 7 × 19 × 18543377 6. 2466269143 = 13 × 173 × 541 × 2027 2466269147 is prime 2466269149 is prime 2466269153 is prime 2466269159 is prime 2466269161 is prime 2466269167 is prime 7. 2466269171 = 43 × 57355097 2466269173 is prime 2466269177 is prime 8. 2466269179 = 11 × 19 × 631 × 18701 9. 2466269183 = 7 × 103 × 113 × 30271 2466269189 is prime 2466269191 is prime 10. 2466269197 = 7 × 157 × 829 × 2707
11	1. 2967301 = 43 × 151 × 457 2. 2967307 = 7 × 109 × 3889 3. 2967311 = 53 × 55987 4. 2967313 = 719 × 4127 2967317 is prime 5. 2967319 = 79 × 37561 2967323 is prime 2967329 is prime 2967331 is prime 2967337 is prime 6. 2967341 = 13 × 228257 2967343 is prime 2967347 is prime 7. 2967349 = 7 × 11 × 89 × 433 2967353 is prime 2967359 is prime 2967361 is prime 8. 2967367 = 13 × 17 × 29 × 463 9. 2967371 = 11 × 269761 2967373 is prime 10. 2967377 = 7 × 73 × 5807 2967379 is prime 2967383 is prime 2967389 is prime 11. 2967391 = 7 × 7 × 23 × 2633 2967397 is prime 2967401 = 17 × 19 × 9187 2967403 is prime 2967407 is prime 2967409 is prime
	1. 98195303 = 23 × 421 × 10141 98195309 is prime 98195311 is prime 2. 98195317 = 11 × 1621 × 5507 3. 98195321 = 7 × 31 × 149 × 3037 98195323 is prime 98195327 is prime 98195329 is prime 98195333 is prime 4. 98195339 = 11 × 977 × 9137 5. 98195341 = 79 × 1 242979 98195347 is prime 98195351 is prime 98195353 is prime 6. 98195357 = 13 × 1511 × 4999 7. 98195359 = 43 × 103 × 22171 8. 98195363 = 7 × 7 × 19 × 29 × 3637 9. 98195369 = 41 × 2395009 98195371 is prime 10. 98195377 = 7 × 947 × 14813 98195381 is prime 11. 98195383 = 11 × 13 × 17 × 31 × 1303 98195387 is prime 98195389 is prime 98195393 is prime 98195399 is prime

In the case of the 29,674^th century, I have included the first decade of the following century because of the remarkable prime richness of the 100 numbers ending with 2,967,410. The eighteen primes in those 100 numbers is not equalled between 113,180 and 341,078,531,679 (that is, greater than 341 billion, or over one hundred thousand times larger in magnitude than the 29,674^th century), and no consecutive multiples of 100 between 200 and 122,853,771,370,899 contain as many as eighteen primes between them. Moreover, as Brian Kehrig noted in correspondence with me, seven rather than eight is the number of digits for which there should be fewest centuries or sequences exceptionally rich in prime numbers, making the sequence from 2,967,310 to 2,967,410 even more exceptional.

The century from 98,195,300 to 98,195,399 is still within the eight-digit gap — after all, it contains eight digits and not more or less — but the eleven composites not divisible by 2, 3, or 5 is the fewest since the 29,674^th century and shows one moving away from the “core” of that gap. There are four centuries between 98,195,400 and 839,296,299 containing only eleven composites not divisible by 2, 3, or 5, but it would be superfluous to include all of them and the 981,954^th century should be reasonably illustrative.

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, ...
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, ...

No century for n = 4 or n = 5 is yet known, and none beyond the second century for n = 7. 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.