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A373401
Least k such that the k-th maximal antirun of prime numbers > 3 has length n. Position of first appearance of n in A027833. The sequence ends if no such antirun exists.
16
1, 2, 4, 6, 10, 8, 69, 40, 24, 46, 41, 21, 140, 82, 131, 210, 50, 199, 35, 30, 248, 192, 277, 185, 458, 1053, 251, 325, 271, 645, 748, 815, 811, 1629, 987, 826, 1967, 423, 1456, 2946, 1109, 406, 1870, 1590, 3681, 2920, 3564, 6423, 1426, 5953, 8345, 12687, 6846
OFFSET
1,2
COMMENTS
The sorted version is A373402.
For this sequence, we define an antirun to be an interval of positions at which consecutive primes differ by at least 3.
EXAMPLE
The maximal antiruns of prime numbers > 3 begin:
5
7 11
13 17
19 23 29
31 37 41
43 47 53 59
61 67 71
73 79 83 89 97 101
103 107
109 113 127 131 137
139 149
151 157 163 167 173 179
The a(n)-th rows are:
5
7 11
19 23 29
43 47 53 59
109 113 127 131 137
73 79 83 89 97 101
2269 2273 2281 2287 2293 2297 2309
1093 1097 1103 1109 1117 1123 1129 1151
463 467 479 487 491 499 503 509 521
For example, (19, 23, 29) is the first maximal antirun of length 3, so a(3) = 4.
MATHEMATICA
t=Length/@Split[Select[Range[4, 100000], PrimeQ], #1+2!=#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]], SubsetQ[t, Range[#]]&];
Table[Position[t, k][[1, 1]], {k, spna[t]}]
CROSSREFS
For composite instead of prime we have A073051.
For runs instead of antiruns we have the triple (4,2,1), firsts of A251092.
For squarefree instead of prime we have A373128, firsts of A373127.
The sorted version is A373402.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
Sequence in context: A243501 A076246 A100426 * A213474 A187333 A321805
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 09 2024
STATUS
approved