“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, making the strange 41st century pattern still unique. This uniqueness extends as far as century patterns have been tabulated: no century repeating the prime pattern from 4,000 to 4,099 exists below 10¹⁷.