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EXAMPLE
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The irregular triangle begins:
n\k 1 2 3 4 5 6 7 ...
1: 1
2: 1
3: 1 1
4: 1 1
5: 1 2
6: 1 2
7: 1 3 2
8: 1 3 2
9: 1 4 4 1
10: 1 4 4 1
11: 1 5 6 2
12: 1 5 6 2
13: 1 6 11 8 2
14: 1 6 11 8 2
15: 1 7 15 14 4
16: 1 7 15 14 4
17: 1 8 19 20 8 1
18: 1 8 19 20 8 1
19: 1 9 27 39 24 5
20: 1 9 27 39 24 5
21: 1 10 33 54 44 16 2
22: 1 10 33 54 44 16 2
23: 1 11 39 69 62 26 2
24: 1 11 39 69 62 26 2
...
The first admissible k-tuples are (blanks within a tuple are here omitted):
n\k 1 2 3 4 ...
1: [0]
2: [0]
3: [0] [0,2]
4: [0] [0,2]
5: [0] [[0,2], [0,4]]
6: [0] [[0,2], [0,4]]
7: [0] [[0,2], [0,4], [0,6]] [[0,2,6], [0,4,6]]
8: [0] [[0,2], [0,4], [0,6]] [[0,2,6], [0,4,6]]
9: [0] [[0,2], [0,4], [0,6], [0,8]] [[0,2,6], [0,2,8], [0,4,6], [0,6,8]] [0,2,6,8]
10: [0] [[0,2], [0,4], [0,6], [0,8]] [[0,2,6], [0,2,8], [0,4,6], [0,6,8]] [0,2,6,8]
...
The first admissible k-tuples for prime k-constellations are:
n\k 1 2 3 4 5 6 ...
1: [0]
2: [0]
3: [0] [0,2]
4: [0] [0,2]
5: [0] [0,2]
6: [0] [0,2]
7: [0] [0,2] [[0,2,6], [0,4,6]]
8: [0] [0,2] [[0,2,6], [0,4,6]]
9: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8]
10: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8]
11: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8]
12: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8]
13: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8] [[0,2,6,8,12],[0,4,6,10,12]]
14: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8] [[0,2,6,8,12],[0,4,6,10,12]]
15: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8] [[0,2,6,8,12],[0,4,6,10,12]]
16: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8] [[0,2,6,8,12],[0,4,6,10,12]]
17: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8] [[0,2,6,8,12],[0,4,6,10,12]] [0,4,6,10,12,16]
18: [0] [0,2] [[0,2,6], [0,4,6]] [0,2,6,8] [[0,2,6,8,12],[0,4,6,10,12]] [0,4,6,10,12,16]
...
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T(7, 3) = 2 because Ieven_n = [0, 2, 4, 6], and the only admissible 3-tuples from this interval are [0, 2, 6] and [0, 4, 6]. For example, [0, 2, 4] is excluded because the set B_3 (mod 3) = {0, 1, 2}, thus #{0, 1, 2} = 3 and (p = 3) - 3 = 0, not > 0.
These two admissible 3-tuples both have diameter 6 and stand for prime 3-constellations for all n >= 7: p, p + 2, p + 6, and p, p + 4, p + 6. One of the Hardy-Littlewood conjectures is that there are in both cases infinitely many such prime triples. For the first members of such triples see A022004 and A022005.
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