An Extended Table of Values of S(n), and Related Comments

The table below lists the sets S(1), S(2), ..., S(20).

One may use this table as follows: looking at n = 7 for example, it follows that ESC(n) is true for every integer n which is congruent to 20, 23, or 26 (mod 27). For instance, ESC(47) is true, since 47 is congruent to 20 (mod 27).

Also, for each n we identify a number g(n), which gives some "good natured" insight into an idea we're using to establish ESC(n) for larger and larger values of n. Thinking of S(1), S(2), S(3), ... as consecutively applied filters, g(n) counts the primes which "escape" or "pass through" the first n filters, among the first 10,000 primes. That is, g(n) is the number of primes (among the first 10,000) which are not congruent to an element of S(1) mod 3, and not congruent to an element of S(2) mod 7, and not congruent to an element of S(3) mod 11, ... , and not congruent to an element of S(n) mod 4n-1.

Values of S(n)
n
The modulus, 4n-1
The set S(n)
g(n)
1
3
{2}
4989
2
7
{3, 5, 6}
2472
3
11
{7, 8, 10}
1718
4
15
{7, 11, 13, 14}
840
5
19
{14, 15, 18}
704
6
23
{7, 10, 11, 15, 17, 19, 20, 21, 22}
403
7
27
{20, 23, 26}
403
8
31
{15, 23, 27, 29, 30}
332
9
35
{23, 26, 31, 32, 34}
332
10
39
{17, 19, 23, 29, 31, 34, 35, 37, 38}
215
11
43
{32, 39, 42}
197
12
47
{11, 15, 22, 23, 30, 31, 35, 39, 41, 43, 44, 45, 46}
144
13
51
{38, 47, 50}
144
14
55
{24, 27, 39, 41, 47, 48, 51, 53, 54}
123
15
59
{18, 23, 39, 44, 47, 54, 55, 56, 58}
94
16
63
{31, 47, 55, 59, 61, 62}
94
17
67
{50, 63, 66}
89
18
71
{23, 31, 34, 35, 47, 53, 55, 59, 62, 63, 65, 67, 68, 69, 70}
71
19
75
{56, 71, 74}
71
20
79
{15, 37, 39, 47, 58, 59, 63, 69, 71, 74, 75, 77, 78}
57

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