Tuesday, 29 August 2023

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 29,674th 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 29,674th 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 29,674th 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 981,954th century should be reasonably illustrative.

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