The problem of “don’t ask” just from fearing no

This evening I went into a terrible temper tantrum when my mother complained that I left cooking rissoles for tonight’s dinner until they were seriously overcooked and smelling.

The tantrum, with familiar phrases (mantras) like:

“get the word “can” out of your head”

“there’s no such word as “can””

and less familiar mantras like

“I used to be able to [remember to come down] but I can’t now”

and gestures like trying to place the word “can” in the rubbish bin in the kitchen [by pointing to the rubbish bin in a terribly agitated manner], became so bad that I even walked out of the house without my mobile phone! Outside I was just as angry and agitated — trying basically to make my mother recognise that I was totally incapable of reminding myself that I had cooking to remember when I was sitting at the very computer where I am writing this post.

My mother was just as upset as I was — and I might note that she has not been well in recent weeks, having mucus-y coughs for the past week or more. It has been difficult for me to appreciate her troubles, but I hope I becoming less bad at it.

After I came back — deliberately knocking softly in order to, as it were, persuade my mother that I had calmed down — I did speak much more calmly about the issue. I enjoy cooking, but my mother had said that I might have to give it up if I forget it time and time again as I have been doing for the past few weeks.

When I look at it logically, I recognise how I was obsessed with watching football and possessed zero desire to do the cooking. I simply wished to watch the football and listen to music as I had been doing for many hours before I went out to walk the dog, but feared my mother’s opinion too much to say “no”. So, I did the cooking as quickly as possible and was quite lucky in fact not to burn the spring onion, celery and mushroom mix that was to be mixed with the mince to make the rissoles. Although I said to Mummy that I would come back when I put the rissoles in the oven, I immediately and totally forgot as soon as I was back settled at the computer watching football. There was not the slightest thought about what was happening inside the oven until the DVD was finished. So unconcerned was I that I was actually surprised that the rissoles had nearly burned over!

Once I came back, realising that I could not keep screaming and screaming outside, and even feeling my vocal cords somewhat weakened, I tried to talk to my mother about what needs to be done to give some hope this will not happen as consistently as it has been of late. I am probably reasonably convinced that there is no hope I can be instructed to remember to come down when I am totally focused on listening to music or on watching football (of course, anything I am watching on the computer would have the same effect as football!). Both of us were extremely upset at the way I expected my mother to be a “lackey” upon whom we can fall back whenever I forget cooking jobs that I do just to please her when my actual desire is to listen to music or watch something on the computer. I actually say I want to cook to appease my mother when I actually want to spend all day doing exactly what I want — which, today, was to watch football.

However, the most notable lesson from tonight’s tempera tantrum was that my consistent refusal to ask my mother whether she would either:

1. remind me by coming up to the study or
2. look after my cooking

out of fear of being told “no” is really, really bad. It might be wise because I know all too well that if my mother said “no” I would most likely go straight into a meltdown as bad as today’s. However, my deliberate silence has meant that my mother and I are both left ignoring the cooking with risks of burning. When I look at it, I thought of the many old English cricketers — for example Walter Brearley or George Louden — who would likely have greatly improved England’s prospects in Australia but were never questioned about touring because MCC was sufficiently aware they had zero possibility of being away from business for so long. My case is different because I have never tried — out of partially justifiable fear of being told “no” — to ask my mother to watch over my cooking. Instead, I assume she will do so as my slave. When in a tantrum, of course, I justify it as a “one-way street” where I can do and get exactly what I want one hundred percent of the time, although logically this “one-way street” is totally selfish and makes everybody else totally immaterial. It is clear to me — whether I cam capable of change or not (and I have zero belief I am) — that I must do something to prevent me forgetting my cooking, or develop some sort of compromise where I do the preparation and my mother agrees to watch over the cooking. In a logical mood these compromises are terribly unfair to my mother, but given my lack of genuine interest in cooking it is unclear what else we can do??

Where there cannot be the maximum number of primes

In a few posts from last year, I discussed pentatrigesimal representation of numbers using all numbers and all letters except O (“O” being a common reading of the number “0” and very similar in shape).

One interesting discovery I made is that whilst 35 consecutive numbers can contain as many as ten primes, if we tabulate divisibility of numbers between each multiple of 35, we find something quite interesting:

k	6n	6n+1	6n+2	6n+3	6n+4	6n+5
1		2		2	3	2
2	2		2		2	3
3	3	2		2		2
4	2	3	2		2
5	5	2	3	2	5	2
6	2		2	3	2
7	7	2	7	2	3	2
8	2		2		2	3
9	3	2		2		2
A	2	3	2	5	2	5
B		2	3	2		2
C	2		2	3	2
D		2		2	3	2
E	2	7	2	7	2	3
F	3	2	5	2	5	2
G	2	3	2		2
H		2	3	2		2
I	2		2	3	2
J		2		2	3	2
K	2	5	2	5	2	3
L	3	2	7	2	7	2
M	2	3	2		2
N		2	3	2		2
P	2		2	3	2
Q	5	2	5	2	3	2
R	2		2		2	3
S	3	2		2		2
T	2	3	2	7	2	7
U		2	3	2		2
V	2	5	2	3	2	5
W		2		2	3	2
X	2		2		2	3
Y	3	2		2		2
Z	2	3	2		2
Possible primes	8	8	8	8	8	8

What we see is that whilst there can be ten primes in a span of thirty-five numbers, there can as shown above never be more than eight primes between consecutive multiples of 35. The first occurrence of eight primes between multiples of 35 is between the pentatrigesimal numbers 4L0 and 4LZ, or in decimal 5,635 to 5,669, whereby 5,639, 5,641, 5,647, 5,651, 5,653, 5,657, 5,659 and 5,669 are all primes.

What I found reading today is that OEIS has actually tabulated the number of possible primes between multiples of base b, as well as the number possible in an arbitrary sequence of length n. I have compared “decades” and “centuries” in all bases up to 35, with sequences serving “centuries” (k00) shaded in grey:

Base		Arbitrary	Multiple	Difference
2		1	1	0
3		2	1	-1
2	4	2	2	0
5		2	2	0
6		2	2	0
7		3	2	-1
8		3	3	0
3	9	4	3	-1
10		4	4	0
11		4	4	0
12		4	4	0
13		5	4	-1
14		5	4	-1
15		5	4	-1
4	16	5	5	0
17		6	5	-1
18		6	6	0
19		6	6	0
20		6	6	0
21		7	5	-2
22		7	7	0
23		7	7	0
24		7	6	-1
5	25	7	7	0
26		7	7	0
27		8	7	-1
28		8	7	-1
29		8	8	0
30		8	7	-1
31		9	8	-1
32		9	9	0
33		10	8	-2
34		10	10	0
35		10	8	-2
6	36	10	10	0
7	49	13	12	-1
8	64	16	16	0
9	81	20	19	-1
10	100	23	23	0
11	121	27	26	-1
12	144	31	30	-1
13	169	37	36	-1
14	196	41	41	0
15	225	46	45	-1
17	289	58	56	-2
18	324	62	62	0
19	361	68	67	-1
20	400	75	75	0
21	441	81	80	-1
22	484	88	87	-1
23	529	95	94	-1
24	576	101	101	0
25	625	109	109	0
26	676	117	116	-1
27	729	124	124	0
28	784	132	131	-1
29	841	141	139	-2
30	900	149	149	0
31	961	158	156	-2

As you can see, the pentatrigesimal case I discovered earlier is not exceptional. For base 21, one already sees that there can never be more than five primes between multiples of 21, but there can be seven primes within arbitrary sequences of 21 numbers.

Nonetheless, the list is revealing because — as it did for me — it gave a false impression that there must be more prime-dense “decades” in pentatrigesimal than that from 4L0 to 4LZ (5,635 to 5,669 in decimal). In cases like this one does need to take some care looking for prime constellations that can actually be easily proved impossible.

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 wiht 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.