Sunday, 8 December 2024

“0-4-0” composite decade sequences and the strange 41st century

Ever since I studied prime tables for the first time, I have always been struck by the factorisations of numbers in the 41st century — slightly above the point at which I have thoroughly memorised my primes:

Factorisations of Numbers Not Divisible by 2 or 5 in 41st Century:

  • 4001 is prime
  • 4003 is prime
  • 4007 is prime
  • 4009 = 19 × 211
  • 4011 = 3 × 7 × 191
  • 4013 is prime
  • 4017 = 3 × 13 × 103
  • 4019 is prime
  • 4021 is prime
  • 4023 = 3 × 3 × 3 × 149
  • 4027 is prime
  • 4029 = 3 × 17 × 79
  • 4031 = 29 × 139
  • 4033 = 37 × 109
  • 4037 = 11 × 367
  • 4039 = 7 × 577
  • 4041 = 3 × 3 × 449
  • 4043 = 13 × 311
  • 4047 = 3 × 19 × 71
  • 4049 is prime
  • 4051 is prime
  • 4053 = 3 × 7 × 193
  • 4057 is prime
  • 4059 = 3 × 3 × 11 × 41
  • 4061 = 31 × 131
  • 4063 = 17 × 239
  • 4067 = 7 × 7 × 83
  • 4069 = 13 × 313
  • 4071 = 3 × 23 × 59
  • 4073 is prime
  • 4077 = 3 × 3 × 3 × 151
  • 4079 is prime
  • 4081 = 7 × 11 × 53
  • 4083 = 3 × 1361
  • 4087 = 61 × 67
  • 4089 = 3 × 29 × 47
  • 4091 is prime
  • 4093 is prime
  • 4097 = 17 × 241
  • 4099 is prime
What is notable is that:
  • the 41st century is actually prime-rich, with fifteen primes
    • vis-à-vis
      • only nine in the 42nd century
      • only eleven each in the 39th and 40th centuries
      • only twelve in the 38th century
  • yet the 4030s and 4060s are the first consecutive decades with k == 1 (mod 3) k such that 10k+1, 10k+3, 10k+7, and 10k+9 are all composite
This peculiar pattern attracted my attention at the time, and continues to do so. It was a while before I actually recognised the pattern found between 4,010 and 4,080 as the peculiarity it was, at least amongst numbers smaller than ten thousand. All odd numbers in the decades with k == 1 (mod 3) are composite, whereas excluding 4,043 all numbers not divisible by 2, 3 or 5 in decades with k ≠≠ 1 (mod 3) are primes. Some years ago, I searched for similar patterns, and found and memorised two others in the ninth and tenth millennia, and one more in the eleventh millennium. Nonetheless, it always interested me to do two things:
  1. make a large list of groups of three decades analogous to the 4050s, 4060s and 4070s
  2. see how many primes are actually found in the centuries where these groups occur
To do this, I expressed any decade with k == 1 (mod 3) in the form 15k. Any decade k0 with k == 1 (mod 3) will have the number k5 divisible by 15, and expressible as 15k where k is odd. One could express it as 15(2k+1) where k is an integer but I prefer to use the “established” expression based upon 15k — the simplest available.

Numbers between neighbouring multiples of 15 and not divisible by 2, 3, or 5 have the formulae:
  1. 15k±14
  2. 15k±8
  3. 15k±4
  4. 15k±2
Analogous sequences to that between 4,050 and 4,080 require that:
  1. all four numbers in 1) and 2) above are prime, but
  2. all four numbers in 3) and 4) are composite
Below are tabulated all such decades in the first three million natural numbers:

ks less than three million such that 15k-14, 15k-8, 15k+8 and 15k+14 are all prime, but 15k-4, 15k-2, 15k+2 and 15k+4 are all composite:

n k p1 p2 p3 p4 Prime count for century/centuries
1 271 4051 4057 4073 4079 15
2 577 8641 8647 8663 8669 13
3 661 9901 9907 9923 9929 9
4 725 10861 10867 10883 10889 10
5 831 12451 12457 12473 12479 13
6 907 13591 13597 13613 13619 8 12
7 2195 32911 32917 32933 32939 13
8 2579 38671 38677 38693 38699 12
9 3195 47911 47917 47933 47939 11
10 3279 49171 49177 49193 49199 12
11 4681 70201 70207 70223 70229 10
12 4939 74071 74077 74093 74099 9
13 5169 77521 77527 77543 77549 13
14 5357 80341 80347 80363 80369 8
15 6661 99901 99907 99923 99929 8
16 7409 111121 111127 111143 111149 9
17 7639 114571 114577 114593 114599 6
18 9713 145681 145687 145703 145709 9 10
19 12037 180541 180547 180563 180569 8
20 14087 211291 211297 211313 211319 11 8
21 14597 218941 218947 218963 218969 9
22 15495 232411 232417 232433 232439 10
23 15991 239851 239857 239873 239879 10
24 16159 242371 242377 242393 242399 7
25 16305 244561 244567 244583 244589 9
26 16455 246811 246817 246833 246839 8
27 17365 260461 260467 260483 260489 10
28 17509 262621 262627 262643 262649 9
29 17589 263821 263827 263843 263849 11
30 18601 279001 279007 279023 279029 6
31 18981 284701 284707 284723 284729 13
32 19833 297481 297487 297503 297509 8 6
33 20071 301051 301057 301073 301079 7
34 20669 310021 310027 310043 310049 8
35 20725 310861 310867 310883 310889 9
36 21163 317431 317437 317453 317459 9
37 22857 342841 342847 342863 342869 10
38 23075 346111 346117 346133 346139 8
39 24937 374041 374047 374063 374069 10
40 25651 384751 384757 384773 384779 8
41 25849 387721 387727 387743 387749 8
42 25883 388231 388237 388253 388259 7
43 26115 391711 391717 391733 391739 8
44 26301 394501 394507 394523 394529 8
45 27625 414361 414367 414383 414389 9
46 31433 471481 471487 471503 471509 7 9
47 31461 471901 471907 471923 471929 9
48 31917 478741 478747 478763 478769 9
49 32471 487051 487057 487073 487079 10
50 32869 493021 493027 493043 493049 11
51 33379 500671 500677 500693 500699 6
52 33847 507691 507697 507713 507719 7 7
53 34543 518131 518137 518153 518159 11
54 35123 526831 526837 526853 526859 6
55 36165 542461 542467 542483 542489 8
56 36895 553411 553417 553433 553439 9
57 38231 573451 573457 573473 573479 10
58 38371 575551 575557 575573 575579 9
59 42313 634681 634687 634703 634709 8 9
60 42563 638431 638437 638453 638459 7
61 42599 638971 638977 638993 638999 7
62 43433 651481 651487 651503 651509 7 4
63 43691 655351 655357 655373 655379 9
64 44083 661231 661237 661253 661259 7
65 47633 714481 714487 714503 714509 5 10
66 49437 741541 741547 741563 741569 7
67 49453 741781 741787 741803 741809 4 10
68 50493 757381 757387 757403 757409 8 8
69 50859 762871 762877 762893 762899 8
70 51371 770551 770557 770573 770579 12
71 52125 781861 781867 781883 781889 9
72 53015 795211 795217 795233 795239 8
73 53399 800971 800977 800993 800999 9
74 55731 835951 835957 835973 835979 11
75 57075 856111 856117 856133 856139 9
76 59243 888631 888637 888653 888659 9
77 59265 888961 888967 888983 888989 9
78 60237 903541 903547 903563 903569 5
79 60505 907561 907567 907583 907589 7
80 61359 920371 920377 920393 920399 7
81 63375 950611 950617 950633 950639 10
82 64285 964261 964267 964283 964289 11
83 65621 984301 984307 984323 984329 13
84 66357 995341 995347 995363 995369 12
85 66393 995881 995887 995903 995909 4 9
86 69289 1039321 1039327 1039343 1039349 7
87 69345 1040161 1040167 1040183 1040189 11
88 70395 1055911 1055917 1055933 1055939 8
89 70681 1060201 1060207 1060223 1060229 8
90 71915 1078711 1078717 1078733 1078739 7
91 72319 1084771 1084777 1084793 1084799 8
92 72693 1090381 1090387 1090403 1090409 5 10
93 73565 1103461 1103467 1103483 1103489 7
94 73881 1108201 1108207 1108223 1108229 8
95 77535 1163011 1163017 1163033 1163039 10
96 78327 1174891 1174897 1174913 1174919 7 6
97 79861 1197901 1197907 1197923 1197929 9
98 80069 1201021 1201027 1201043 1201049 11
99 81429 1221421 1221427 1221443 1221449 8
100 82079 1231171 1231177 1231193 1231199 7
101 82353 1235281 1235287 1235303 1235309 7 8
102 82771 1241551 1241557 1241573 1241579 8
103 83645 1254661 1254667 1254683 1254689 11
104 84991 1274851 1274857 1274873 1274879 6
105 86089 1291321 1291327 1291343 1291349 10
106 88239 1323571 1323577 1323593 1323599 9
107 88259 1323871 1323877 1323893 1323899 6
108 89045 1335661 1335667 1335683 1335689 9
109 89231 1338451 1338457 1338473 1338479 9
110 91647 1374691 1374697 1374713 1374719 9 8
111 91733 1375981 1375987 1376003 1376009 7 6
112 92339 1385071 1385077 1385093 1385099 11
113 93299 1399471 1399477 1399493 1399499 10
114 94749 1421221 1421227 1421243 1421249 8
115 94769 1421521 1421527 1421543 1421549 6
116 95847 1437691 1437697 1437713 1437719 8 7
117 96273 1444081 1444087 1444103 1444109 6 5
118 97827 1467391 1467397 1467413 1467419 9 6
119 99377 1490641 1490647 1490663 1490669 10
120 101047 1515691 1515697 1515713 1515719 6 9
121 102373 1535581 1535587 1535603 1535609 7 8
122 104613 1569181 1569187 1569203 1569209 9 7
123 104655 1569811 1569817 1569833 1569839 7
124 107669 1615021 1615027 1615043 1615049 8
125 108301 1624501 1624507 1624523 1624529 7
126 108727 1630891 1630897 1630913 1630919 8 6
127 109175 1637611 1637617 1637633 1637639 9
128 111675 1675111 1675117 1675133 1675139 8
129 112369 1685521 1685527 1685543 1685549 9
130 115747 1736191 1736197 1736213 1736219 9 8
131 116363 1745431 1745437 1745453 1745459 8
132 118393 1775881 1775887 1775903 1775909 7 6
133 118407 1776091 1776097 1776113 1776119 8 7
134 119779 1796671 1796677 1796693 1796699 6
135 119927 1798891 1798897 1798913 1798919 7 10
136 121371 1820551 1820557 1820573 1820579 9
137 122253 1833781 1833787 1833803 1833809 9 6
138 122561 1838401 1838407 1838423 1838429 6
139 123471 1852051 1852057 1852073 1852079 10
140 123797 1856941 1856947 1856963 1856969 11
141 123967 1859491 1859497 1859513 1859519 8 8
142 124217 1863241 1863247 1863263 1863269 7
143 125477 1882141 1882147 1882163 1882169 7
144 127957 1919341 1919347 1919363 1919369 8
145 128103 1921531 1921537 1921553 1921559 6
146 129713 1945681 1945687 1945703 1945709 10 10
147 130765 1961461 1961467 1961483 1961489 10
148 131665 1974961 1974967 1974983 1974989 7
149 134249 2013721 2013727 2013743 2013749 12
150 135615 2034211 2034217 2034233 2034239 8
151 136355 2045311 2045317 2045333 2045339 10
152 137259 2058871 2058877 2058893 2058899 12
153 139165 2087461 2087467 2087483 2087489 8
154 140191 2102851 2102857 2102873 2102879 10
155 140949 2114221 2114227 2114243 2114249 8
156 141703 2125531 2125537 2125553 2125559 8
157 142329 2134921 2134927 2134943 2134949 7
158 142999 2144971 2144977 2144993 2144999 6
159 145953 2189281 2189287 2189303 2189309 9 11
160 146613 2199181 2199187 2199203 2199209 9 8
161 149215 2238211 2238217 2238233 2238239 9
162 149531 2242951 2242957 2242973 2242979 8
163 151099 2266471 2266477 2266493 2266499 9
164 151609 2274121 2274127 2274143 2274149 7
165 153193 2297881 2297887 2297903 2297909 8 8
166 153967 2309491 2309497 2309513 2309519 7 5
167 154157 2312341 2312347 2312363 2312369 6
168 155675 2335111 2335117 2335133 2335139 8
169 157939 2369071 2369077 2369093 2369099 6
170 159301 2389501 2389507 2389523 2389529 10
171 159717 2395741 2395747 2395763 2395769 7
172 161481 2422201 2422207 2422223 2422229 10
173 166583 2498731 2498737 2498753 2498759 8
174 169223 2538331 2538337 2538353 2538359 11
175 169229 2538421 2538427 2538443 2538449 11
176 169983 2549731 2549737 2549753 2549759 7
177 172063 2580931 2580937 2580953 2580959 8
178 173239 2598571 2598577 2598593 2598599 10
179 173365 2600461 2600467 2600483 2600489 7
180 175499 2632471 2632477 2632493 2632499 8
181 175541 2633101 2633107 2633123 2633129 7
182 175863 2637931 2637937 2637953 2637959 9
183 176369 2645521 2645527 2645543 2645549 12
184 176545 2648161 2648167 2648183 2648189 8
185 179659 2694871 2694877 2694893 2694899 11
186 180269 2704021 2704027 2704043 2704049 6
187 180911 2713651 2713657 2713673 2713679 7
188 182753 2741281 2741287 2741303 2741309 10 6
189 184767 2771491 2771497 2771513 2771519 6 7
190 186363 2795431 2795437 2795453 2795459 10
191 187577 2813641 2813647 2813663 2813669 9
192 187953 2819281 2819287 2819303 2819309 6 7
193 189249 2838721 2838727 2838743 2838749 7
194 189745 2846161 2846167 2846183 2846189 7
195 194899 2923471 2923477 2923493 2923499 10
196 194917 2923741 2923747 2923763 2923769 7
197 196087 2941291 2941297 2941313 2941319 8 7
198 196149 2942221 2942227 2942243 2942249 10
199 197117 2956741 2956747 2956763 2956769 8
200 197335 2960011 2960017 2960033 2960039 7
201 199937 2999041 2999047 2999063 2999069 6
The table above shows two hundred and one cases of the pattern discussed above for the natural numbers between 4,050 and 4,080 (and with one exception between 4,010 and 4,050). Nevertheless, there are none between 4,080 and three million within a century containing fourteen primes, let alone fifteen. Nor are there any within the twenty-four centuries with fourteen or fifteen primes in the next seven millions. There is a gap of sixteen between the primes 10,014,427 and 10,014,443 in the first (of fifty) eight-digit centuries with fourteen primes. Nevertheless, the first fourteen-prime century containing a sequence of the type tabulated above occurs between 96,377,200 and 96,377,299 — the 114th century containing fourteen prime numbers. (The 41st century is the eighth with fifteen primes).

Given that 4,043 is divides by thirteen and not seven, there should exist a possibility of two consecutive odd k in the table above, but there exist none less than three million, although fifty-six cases of gaps of sixteen following two consecutive numbers of the form 15k-8 occur in that range. There is even one case of gaps of sixteen after three successive (prime) numbers of form 15k-8 — after the primes 2,492,767, 2,492,797 and 2,492,827. Again, though, this sequence covers two seven-prime centuries.

The strange 41st century pattern thus remains unique. It stands unique as far as century patterns have been tabulated: no century repeating the prime pattern from 4,000 to 4,099 exists below minimally 1017.
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Friday, 22 November 2024

“Full digits” in sixteen-prime centuries

In my previous post, I noted that I was trying to see how many of the smallest sixteen-prime centuries had seven primes ending in one digit. This latter phenomenon is one I have only noticed recently and is not recorded on OEIS at all.

Working through the first one thousand centuries with sixteen primes on OEIS (sequence A186408) I found the following cases:

nth century with 16 primes k n mod 21 Seven primes ending in
23 1097546872 16 9
223 926471559529 19 3
255 1194384398740 1 7
289 1534136303215 19 3
294 1553991849772 19 3
307 1644652079911 1 7
315 1725961694479 1 7
482 3627907603807 4 1
750 8673617675503 4 1
883 11557194277084 1 7
925 12714434652190 19 3
The table shows more cases of seven primes ending in 3 or 7 than on 1 or 9. This is because, as noted in the previous post, there is a very strong preponderance of sixteen-prime centuries that are either 1 or 19 modulo 21. These two moduli 21 account for over half of the first thousand sixteen-prime centuries above the eight-digit gap. As you can see, a 1 modulo 21 century may have seven primes ending in 7 (but no other digit) and a 19 modulo 21 century may have seven primes ending in 3 but no other digit. A couple of year ago I calculated that the probability of a 16-prime century having seven primes ending in the same digit was about 1 in 100. The eleven centuries above agrees fairly well with this expectation.

I have not checked seventeen- or eighteen- prime centuries — this is tough to do because Excel cannot read numbers beyond 1015 to the precise whole number — but the fact that the first seventeen-prime century with seven primes ending in one digit has seven primes ending in 7 does suggest a similar pattern.

The full factorisations of the smallest century with seven primes ending in each digit, with primes formign the group of seven coloured in red:
  • 109754687201 = 641 × 1249 × 137089
  • 109754687203 = 7 × 439 × 35 715811
  • 109754687207 = 11 × 17 × 83 × 7071367
  • 109754687209 is prime
  • 109754687213 is prime
  • 109754687219 is prime
  • 109754687221 is prime
  • 109754687227 = 499 × 947 × 232259
  • 109754687231 = 7 × 41 × 827 × 462419
  • 109754687233 is prime
  • 109754687237 = 13 × 89 × 97 × 157 × 6229
  • 109754687239 is prime
  • 109754687243 is prime
  • 109754687249 is prime
  • 109754687251 = 113 × 82460321
  • 109754687257 is prime
  • 109754687261 is prime
  • 109754687263 = 13 × 37 × 228180223
  • 109754687267 is prime
  • 109754687269 is prime
  • 109754687273 = 74 × 11 × 31 × 134053
  • 109754687279 is prime
  • 109754687281 = 29 × 79 × 47906891
  • 109754687287 = 7 × 67 × 107 × 239 × 9151
  • 109754687291 = 19 × 73 × 79130993
  • 109754687293 is prime
  • 109754687297 is prime
  • 109754687299 is prime
  • 92647155952901 = 11 × 673 × 1229 × 10182923
  • 92647155952903 is prime
  • 92647155952907 is prime
  • 92647155952909 is prime
  • 92647155952913 is prime
  • 92647155952919 = 41 × 3167 × 713510177
  • 92647155952921 = 17 × 863 × 48397 × 130483
  • 92647155952927 is prime
  • 92647155952931 = 24953 × 3712866427
  • 92647155952933 is prime
  • 92647155952937 is prime
  • 92647155952939 = 7 × 73 × 181305588949
  • 92647155952943 is prime
  • 92647155952949 = 13 × 79 × 32099 × 2810413
  • 92647155952951 = 29 × 229 × 13950783911
  • 92647155952957 = 186187 × 497602711
  • 92647155952961 is prime
  • 92647155952963 is prime
  • 92647155952967 = 7 × 11 × 79843 × 15069697
  • 92647155952969 is prime
  • 92647155952973 is prime
  • 92647155952979 is prime
  • 92647155952981 = 7 × 23 × 575448173621
  • 92647155952987 is prime
  • 92647155952991 = 19 × 97 × 50269753637
  • 92647155952993 is prime
  • 92647155952997 is prime 
  • 92647155952999 = 53 × 647 × 10133 × 266633
  • 119438439874001 = 19 × 2425019 × 2592241
  • 119438439874003 = 29 × 4118566892207 
  • 119438439874007 is prime
  • 119438439874009 is prime
  • 119438439874013 = 13 × 53 × 79 × 1787 × 1227929
  • 119438439874019 = 7 × 1907 × 13687 × 653713
  • 119438439874021 is prime 
  • 119438439874027 is prime
  • 119438439874031 = 23 × 257 × 20206130921
  • 119438439874033 = 7 × 161233 × 105825943
  • 119438439874037 is prime
  • 119438439874039 = 11 × 13 × 19 × 149 × 211 × 613 × 2281
  • 119438439874043 = 1473853 × 81038231
  • 119438439874049 is prime
  • 119438439874051 is prime
  • 119438439874057 is prime
  • 119438439874061 = 7 × 11 × 29 × 97 × 6863 × 80347
  • 119438439874063 is prime
  • 119438439874067 is prime
  • 119438439874069 is prime
  • 119438439874073 = 1221083 × 97813531
  • 119438439874079 is prime
  • 119438439874081 is prime
  • 119438439874087 is prime
  • 119438439874091 = 13 × 37 × 59 × 18341 × 229469
  • 119438439874093 = 17 × 89459 × 78536431
  • 119438439874097 is prime
  • 119438439874099 is prime
  • 362790760380701 is prime
  • 362790760380703 is prime
  • 362790760380707 = 17 × 19 × 109 × 10304506501
  • 362790760380709 is prime
  • 362790760380713 = 7 × 51 827251 482959
  • 362790760380719 = 11 × 32980978216429
  • 362790760380721 is prime
  • 362790760380727 = 7 × 31 × 193 × 16889 × 512903
  • 362790760380731 is prime
  • 362790760380733 = 1759 × 206248300387
  • 362790760380737 = 13 × 43 × 53 × 71 × 172 468661
  • 362790760380739 is prime
  • 362790760380743 = 433 × 837853950071
  • 362790760380749 is prime
  • 362790760380751 is prime
  • 362790760380757 is prime
  • 362790760380761 is prime
  • 362790760380763 = 11 × 13 × 13 × 59 × 641 × 947 × 5449
  • 362790760380767 is prime
  • 362790760380769 = 7 × 23 × 47 × 401 × 557 × 214651
  • 362790760380773 is prime
  • 362790760380779 = 37 × 61 × 509 × 11783 × 26801
  • 362790760380781 is prime
  • 362790760380787 is prime
  • 362790760380791 is prime
  • 362790760380793 is prime
  • 362790760380797 = 7 × 51827251482971
  • 362790760380799 = 41 × 631 × 14023066769
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Revealing the eight-digit gap again

In previous posts on prime numbers, I have noticed the existence of an “eight-digit gap” — a range of magnitudes within which the maximum number of primes in a century reaches a minus, and extending from about three million to 840 million.

Today, I attempted to see how many of the first 1,000 centuries with sixteen prime numbers, nine of which lie below the eight-digit gap, contained seven primes ending in one digit. As noted here, the first centuries with seven primes ending in same digit lie in the absolute core of the eight-digit gap, but I wanted to see just how common they are among sixteen-prime centuries larger than that gap. I previously did very brief studies for seventeen- and eighteen-prime centuries, and found that the sixteenth century with eighteen primes from 140,326,343,186,616,700 to 140,326,343,186,616,799 had seven primes ending in 1, but not seventeen-prime century had seven primes ending in any digit until the 179th such century from 24,738,663,087,001,600 to 24,738,663,087,001,699 with seven primes ending in 7.

However, when working out possible cases of seven primes ending in the same digit, I noticed more clearly that the frequency of moduli 21 of sixteen-prime centuries larger than the eight-digit gap was in no way random. (Moduli 21 can exclude any century having seven primes ending in all or all but one of the four digits in which a multi-digit prime may terminate).

Number of First 1,000 Sixteen-Prime Centuries n modulo 21:

All centuries Centuries above eight-digit gap Centuries below eight-digit gap (includes 17- and 21-prime centuries)
n mod 21 total percent n mod 21 total percent n mod 21 total percent
0 4 0.40% 0 3 0.30% 0 1 7.14%
1 276 27.60% 1 276 27.85% 1 2 14.29%
2 9 0.90% 2 8 0.81% 2 1 7.14%
3 16 1.60% 3 15 1.51% 3 1 7.14%
4 58 5.80% 4 58 5.85% 4 2 14.29%
5 7 0.70% 5 6 0.61% 5 1 7.14%
6 20 2.00% 6 19 1.92% 6 1 7.14%
7 28 2.80% 7 28 2.83% 7 0 0.00%
8 33 3.30% 8 33 3.33% 8 0 0.00%
9 7 0.70% 9 7 0.71% 9 0 0.00%
10 68 6.80% 10 66 6.66% 10 2 14.29%
11 14 1.40% 11 14 1.41% 11 0 0.00%
12 27 2.70% 12 27 2.72% 12 0 0.00%
13 33 3.30% 13 33 3.33% 13 0 0.00%
14 31 3.10% 14 31 3.13% 14 1 7.14%
15 8 0.80% 15 8 0.81% 15 0 0.00%
16 48 4.80% 16 47 4.74% 16 1 7.14%
17 11 1.10% 17 11 1.11% 17 0 0.00%
18 9 0.90% 18 9 0.91% 18 0 0.00%
19 282 28.20% 19 282 28.46% 19 0 0.00%
20 11 1.10% 20 10 1.01% 20 1 7.14%
What the table shows is that:
  1. above the eight-digit gap there is a systematic variation in the frequencies of moduli 21 amongst sixteen-prime centuries, with extreme cases shaded
  2. the fourteen centuries below the eight-digit gap with sixteen or more primes do not appear to follow this pattern
  3. this suggests that there is a systematic character to prime-dense centuries above the eight-digit gap absent below that range of magnitudes
There is logic behind this pattern in that small prime factors have to combine in a few numbers with many factors to produce prime-rich centuries above (and within) the eight-digit gap. (As I note here, they also have to combine to produce large prime-free sequences, and this becomes harder as numbers get bigger because it is difficult for enough factors to “synchronise”). Below the eight-digit gap, there are fewer possible factors so there is less need to combine, and thus prime-rich centuries are less likely to fit into repetitive large groups of primes.
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