Four sub-sequences revealed

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

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

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

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

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

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

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

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

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

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

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

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