Some notable fourteen-prime centuries

Today, I had a look for “full digits” in fourteen-and fifteen-prime centuries between the beginning of the “eight-digit gap” and 10¹¹ (100 billion or 100,000,000).

Given my finding that the 1,790,288^th century — overall the thirty-seventh century with exactly fifteen prime numbers — contained seven primes ending in 9, it was only natural that I would seek to search for other fifteen- and fourteen-prime centuries with seven primes ending in one digit. [The 80,563^rd and 768,053^rd centuries with thirteen primes I already knew as having seven primes ending in one digit, so that count did not need discussion. Smaller counts would be too unlikely to have so many primes ending in a single digit, although because centuries with fewer primes are so much more numerous there are actually more “full”-digit centuries with fewer than thirteen total primes].

Early checks of my sequences of centuries with seven primes ending in one digit versus OEIS sequence A186406 did not find a case where there were fourteen total primes and seven ending in one digit as far as I could search. I did another check as far as I could go using the method of modulo(3003) discussed here and checked to see if further centuries in A186406 had a potential “full” century. After I finished my work I thought that I ought to check the fifteen-prime centuries I within the same range of magnitudes.

First 14-Prime century with Empty Digit: “Empty-1” 1,015,316^th:

101531501 = 229 × 443369
1. 101531503 is prime [1.]
101531507 = 7 × 11 × 59 × 22349
1. 101531509 is prime [1.]
2. 101531513 is prime [2.]
3. 101531519 is prime [2.]
101531521 = 7 × 13 × 1115731
4. 101531527 is prime [1.]
101531531 = 17 × 5972443
5. 101531533 is prime [3.]
6. 101531537 is prime [2.]
101531539 = 41 × 73 × 33923
7. 101531543 is prime [4.]
101531549 = 7 × 97 × 149531
101531551 = 11 × 107 × 86263
101531557 = 43 × 2361199
101531561 = 9133 × 11117
101531563 = 7 × 101 × 143609
8. 101531567 is prime [3.]
9. 101531569 is prime [3.]
101531573 = 11 × 13 × 19 × 37369
10. 101531579 is prime [4.]
101531581 = 29 × 3501089
11. 101531587 is prime [4.]
101531591 = 7 × 23 × 199 × 3169
12. 101531593 is prime [5.]
13. 101531597 is prime [5.]
101531599 = 13 × 17 × 67 × 6857

This century is a little strange given that, as the first fourteen-prime century with no primes ending in one of 1, 3, 7, or 9, it is the 116^th century with fourteen primes — those easy to remember in a table because it is the smallest with nine rather than eight digits. The century from 252,724,900 to 252,724,999 with fifteen primes but none ending in 7, is the fortieth century with fifteen primes. The first thirteen-prime century with no primes ending in one of 1, 3, 7, or 9 is the century from 63,600 to 63,699 (no primes ending in 3; overall the 36^th century with thirteen primes), and the first such twelve-prime century is that from 16,400 to 16,499 (no primes ending in 9; overall the 39^th century with twelve primes).

The nine centuries beginning with 200 produce via deletion three clear fourteen-prime patterns with no primes ending in one digit:

1. {7, 13, 17, 37, 47, 49, 53, 59, 67, 73, 79, 83, 89, 97} via deletion from the fourth century has fourteen primes with none ending in 1
2. {1, 7, 13, 17, 31, 41, 43, 47, 53, 61, 73, 77, 83, 91} via deletion from the seventh century has fourteen primes with none ending in 9
3. {9, 13, 19, 21, 31, 33, 39, 49, 51, 61, 63, 69, 91, 93} via deletion from the eleventh century has fourteen primes with none ending in 7

Contrariwise, no fifteen-prime pattern with no primes ending in one digit can be created via deletion from any century between 200 and 252,724,899.

The Only “Full Digit” 14-Prime Century in First Thousand: “Full 7” 113,684,594^th:

1. 11368459301 is prime
11368459303 = 101 × 139 × 263 × 3079
2. 11368459307 is prime [1.]
11368459309 = 37 × 59 × 5207723
3. 11368459313 is prime
11368459319 = 7 × 811 × 2002547
4. 11368459321 is prime
5. 11368459327 is prime [2.]
11368459331 = 29 × 2243 × 174773
11368459333 = 7 × 11 × 157 × 487 × 1931
6. 11368459337 is prime [3.]
11368459339 = 2633 × 4317683
11368459343 = 23 × 233 × 2121377
11368459349 = 13 × 53 × 673 × 24517
11368459351 = 17 × 668732903
7. 11368459357 is prime [4.]
11368459361 = 7 × 13921 × 116663
8. 11368459363 is prime
9. 11368459367 is prime [5.]
11368459369 = 167 × 2333 × 29179
11368459373 = 19 × 598339967
10. 11368459379 is prime
11. 11368459381 is prime
12. 11368459387 is prime [6.]
11368459391 = 1523 × 7464517
13. 11368459393 is prime
14. 11368459397 is prime [7.]
11368459399 = 11 × 7879 × 131171

This is the only case amongst the first thousand fourteen-prime centuries where seven of the primes end in one digit. It is the 674^th century with exactly fourteen primes. It is surprising that this should be so when one sees that the “full 1” 80,563^rd century is only the 263^rd century with thirteen primes and the “full 3” 768,053^rd the 568^th. The surprise is even greater since a fourteen-prime century must logically have greater likelihood of a “full” digit than a thirteen-prime century: I recall that a thirteen-prime century had something like a 1-in-490 chance of a “full” digit and a fourteen-prime century closer to 1-in-340.

130,724,018^th century: 14 primes with gap of 36:

1. 13072401701 is prime
2. 13072401703 is prime
3. 13072401707 is prime
4. 13072401709 is prime
13072401713 = 7 × 12329 × 151471
13072401719 = 13 × 17 × 109 × 127 × 4273
5. 13072401721 is prime
13072401727 = 7 × 11 × 463 × 366677
6. 13072401731 is prime
7. 13072401733 is prime
8. 13072401737 is prime
13072401739 = 23 × 199 × 2856107
9.13072401743 is prime
13072401749 = 11 × 31 × 43 × 891523
10. 13072401751 is prime
11. 13072401757 is prime
12. 13072401761 is prime
13. 13072401763 is prime
13072401767 = 137 × 95418991
13072401769 = 7 × 9241 × 202087
13072401773 = 6379 × 2049287
13072401779 = 57787 × 226217
13072401781 = 73 × 179073997
13072401787 = 17 × 768964811
13072401791 = 89 × 331 × 443749
13072401793 = 11 × 19 × 577 × 108401
13072401797 = 7 × 13 × 143652767
14. 13072401799 is prime

Here we see a century with fourteen primes but a gap of thirty-six, which surprised me when I had a look, although it is less so when one studies the 240,728,320,643^rd century, where by deleting one could have fifteen primes with a gap of 40 in the middle, or much more “familiarly”, the amazing century from 15,640 to 15,740 with fourteen primes, two prime quadruples at the ends, and a record gap of 43 consecutive composite numbers between 15,683 and 15,727 in the middle.

The 219,562,919^th Century: A Prime Decaplet

21956291801 = 59 × 372140539
21956291803 = 13807 × 1590229
1. 21956291807 is prime
21956291809 = 17 × 23 × 56154199
2. 21956291813 is prime
21956291819 = 7 × 11 × 29 × 9832643
21956291821 = 53 × 414269657
3. 21956291827 is prime
21956291831 = 353 × 1693 × 36739
21956291833 = 7 × 19 × 19 × 41 × 73 × 2903
21956291837 = 71 × 397 × 778951
21956291839 = 349 × 62912011
21956291843 = 17 × 953 × 1355243
21956291849 = 31 × 6337 × 111767
21956291851 = 13 × 163 × 1831 × 5659
4. 21956291857 is prime
21956291861 = 7 × 7 × 448087589
21956291863 = 11 × 541 × 3689513
5. 21956291867 is prime
6. 21956291869 is prime
7. 21956291873 is prime
8. 21956291879 is prime
9. 21956291881 is prime
10. 21956291887 is prime
11. 21956291891 is prime
12. 21956291893 is prime
13. 21956291897 is prime
14. 21956291899 is prime

Here we see a fourteen-prime century ending with a prime decaplet — a group of ten primes in 33 integers, which is the shortest span in which ten consecutive primes can occur. OEIS sequences A027569 and A027570 show it as the third prime decaplet after those beginning with 11 and 9853497737. I had never seen anything like this, and checking a few other “small” decaplets from OEIS shows that most are in centuries with even fewer “extraneous” primes than the four in the century above.

The 819,751,072nd century — Eight Primes in 37 at End of Century:

81975107101 = 401 × 967 × 211403
1. 81975107107 is prime
2. 81975107111 is prime
3. 81975107113 is prime
81975107117 = 811 × 6053 × 16699
81975107119 = 83 × 509 × 1940377
81975107123 = 7 × 13 × 4637 × 194269
81975107129 = 73 × 461 × 2435893
81975107131 = 6947 × 11800073
81975107137 = 7 × 11 × 19 × 41 × 1366639
4. 81975107141 is prime
5. 81975107143 is prime
6. 81975107147 is prime
81975107149 = 13 × 39371 × 160163
81975107153 = 49871 × 1643743
81975107159 = 11 × 17 × 3767 × 116371
7. 81975107161 is prime
8. 81975107167 is prime
9. 81975107171 is prime
10. 81975107173 is prime
11. 81975107177 is prime
81975107179 = 7 × 7 × 107 × 15635153

12. 81975107183 is prime
13. 81975107189 is prime
14. 81975107191 is prime
15. 81975107197 is prime

This is a fifteen-prime century, but still quite interesting for the similarity to the previous section in the massive cluster at the end. What is actually more notable here is that although there are only eleven composite numbers coprime to 30, six of them for decades with no primes.