Are prime-poor small centuries “expectedly” so?

On Mersenne Forum, there was a recent response to my previous blog post about moduli (3003), showing that not all centuries that are exceptionally poor in prime numbers have unusually few “potential” primes [“potential” primes being defined as numbers not divisible by 2, 3, 5, 7, 11, or 13 according to modulus (3003), with 3003 being the product of the first four primes coprime to 100].

In order to see whether centuries with abnormally few primes actually have few “potential” primes, I have aimed to test the modulus (3003) of every century up to ten million that has a record-low or equal-record-low number of primes (compared to smaller centuries only of course). These centuries, with their modulus (3003) and “potential” primes, are tabulated below

k	Total primes	mod (3003)	Total possible primes	...1	...3	...7	...9
2	16	2	17	5	5	3	4
3	16	3	19	4	5	6	4
5	14	5	18	4	4	5	5
7	14	7	19	4	5	4	6
9	14	9	18	6	3	6	3
11	12	11	17	4	5	3	5
13	11	13	18	4	5	5	4
21	10	21	15	4	5	3	3
24	10	24	18	4	3	6	5
31	10	31	20	6	4	5	5
41	9	41	17	4	4	5	4
43	9	43	21	4	5	6	6
48	8	48	19	7	4	5	3
59	7	59	19	3	6	5	5
95	7	95	18	4	5	5	4
142	7	142	20	4	5	5	6
165	7	165	18	5	3	5	5
167	7	167	19	5	3	5	6
186	6	186	17	5	3	5	4
188	5	188	17	3	4	5	5
273	5	273	20	6	5	5	4
314	4	314	17	3	5	4	5
356	4	356	19	5	4	6	4
524	4	524	18	4	4	6	4
588	3	588	19	5	5	4	5
695	3	695	17	5	4	4	4
797	3	797	16	4	3	5	4
1430	3	1430	17	5	5	3	4
1559	1	1559	17	4	5	4	4
2683	1	2683	20	5	4	4	7
4133	1	1130	17	5	6	3	3
10048	1	1039	20	5	6	5	4
11400	1	2391	18	5	3	5	5
12727	1	715	21	5	5	7	4
12800	1	788	19	3	6	5	5
13572	1	1560	21	6	4	6	5
14223	1	2211	19	6	4	4	5
14443	1	2431	19	5	3	6	5
14514	1	2502	16	3	4	5	4
14680	1	2668	21	5	5	6	5
14913	1	2901	19	4	4	6	5
15536	1	521	19	4	7	4	4
15619	1	604	20	5	4	4	7
16538	1	1523	17	4	6	3	4
16557	1	1542	18	5	4	5	4
16718	0	1703	20	6	6	3	5
26378	0	2354	17	5	5	3	4
31173	0	1143	17	4	4	5	4
39336	0	297	18	3	4	6	5
46406	0	1361	18	5	5	4	4
46524	0	1479	17	4	3	6	4
51782	0	731	17	4	6	3	4
55187	0	1133	16	3	4	4	5
58374	0	1317	20	5	6	6	3
58452	0	1395	19	6	4	6	3
60129	0	69	20	6	5	6	3
60850	0	790	19	5	5	4	5
63338	0	275	16	4	5	2	5
63762	0	699	19	5	4	6	4
67898	0	1832	17	3	5	4	5
69587	0	518	20	5	5	3	7
71299	0	2230	20	6	4	4	6
75652	0	577	20	5	5	5	5
78035	0	2960	20	4	5	5	6
78269	0	191	19	5	5	4	5
80277	0	2199	18	5	4	5	4
83674	0	2593	19	5	4	5	5
84213	0	129	19	3	5	6	5
89052	0	1965	19	6	4	4	5
95490	0	2397	17	5	4	5	3
97080	0	984	18	5	3	5	5

Results:

If we consider all centuries up to ten million that have an (equal) record low number of primes vis-à-vis preceding centuries, we find that:

“Potential primes”	All centuries	%	“Record few” centuries	%	Difference in %	Difference in ratio
15	4	0.13%	1	1.41%	1.28%	1057.39%
16	46	1.53%	4	5.63%	4.10%	367.79%
17	244	8.13%	17	23.94%	15.82%	294.68%
18	580	19.31%	13	18.31%	-1.00%	94.80%
19	954	31.77%	19	26.76%	-5.01%	84.24%
20	725	24.14%	13	18.31%	-5.83%	75.84%
21	336	11.19%	4	5.63%	-5.56%	50.35%
22	88	2.93%	0	0.00%	-2.93%	0.00%
23	26	0.87%	0	0.00%	-0.87%	0.00%

The table does seem to indicate systematic differences between centuries with the fewest primes up to that point, and all centuries, in terms of the number of potential primes. Whilst it is true that the proportion of moduli (3003) yielding 22 or 23 “potential primes” is very small, it is revealing that no century up to ten million with a record low number of primes has so many. With 19, 20 and 21 “potential primes”, the difference appears to be just as systematic inasmuch as the proportion of centuries with a record small number of primes having so many is consistently fewer than of all centuries. Contrariwise, for 15, 16 and 17 “potential primes”, the proportion is consistently higher for centuries with record few primes than for all centuries. This does imply that centuries with a record small number of primes do have a distinct tendency to have fewer “potential primes” than other centuries, even if the tendency is not consistent enough to apply to every such century.