A373401 - OEIS

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.

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

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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.

LINKS

Table of n, a(n) for n=1..53.

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

EXAMPLE

The maximal antiruns of prime numbers > 3 begin:

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:

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.

Cf. A001359, A005117, A027833, A006512, A053797, A373199, A373405.

Sequence in context: A243501 A076246 A100426 * A213474 A187333 A321805

Adjacent sequences: A373398 A373399 A373400 * A373402 A373403 A373404

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 09 2024

STATUS

approved