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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.

%I A373401 #13 Jun 10 2024 23:36:01
%S A373401 1,2,4,6,10,8,69,40,24,46,41,21,140,82,131,210,50,199,35,30,248,192,
%T A373401 277,185,458,1053,251,325,271,645,748,815,811,1629,987,826,1967,423,
%U A373401 1456,2946,1109,406,1870,1590,3681,2920,3564,6423,1426,5953,8345,12687,6846
%N 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.
%C A373401 The sorted version is A373402.
%C A373401 For this sequence, we define an antirun to be an interval of positions at which consecutive primes differ by at least 3.
%H A373401 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.
%e A373401 The maximal antiruns of prime numbers > 3 begin:
%e A373401     5
%e A373401     7  11
%e A373401    13  17
%e A373401    19  23  29
%e A373401    31  37  41
%e A373401    43  47  53  59
%e A373401    61  67  71
%e A373401    73  79  83  89  97 101
%e A373401   103 107
%e A373401   109 113 127 131 137
%e A373401   139 149
%e A373401   151 157 163 167 173 179
%e A373401 The a(n)-th rows are:
%e A373401      5
%e A373401      7   11
%e A373401     19   23   29
%e A373401     43   47   53   59
%e A373401    109  113  127  131  137
%e A373401     73   79   83   89   97  101
%e A373401   2269 2273 2281 2287 2293 2297 2309
%e A373401   1093 1097 1103 1109 1117 1123 1129 1151
%e A373401    463  467  479  487  491  499  503  509  521
%e A373401 For example, (19, 23, 29) is the first maximal antirun of length 3, so a(3) = 4.
%t A373401 t=Length/@Split[Select[Range[4,100000],PrimeQ],#1+2!=#2&]//Most;
%t A373401 spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#]]&];
%t A373401 Table[Position[t,k][[1,1]],{k,spna[t]}]
%Y A373401 For composite instead of prime we have A073051.
%Y A373401 For runs instead of antiruns we have the triple (4,2,1), firsts of A251092.
%Y A373401 For squarefree instead of prime we have A373128, firsts of A373127.
%Y A373401 The sorted version is A373402.
%Y A373401 A000040 lists the primes, differences A001223.
%Y A373401 A002808 lists the composite numbers, differences A073783.
%Y A373401 A046933 counts composite numbers between primes.
%Y A373401 Cf. A001359, A005117, A027833, A006512, A053797, A373199, A373405.
%K A373401 nonn
%O A373401 1,2
%A A373401 _Gus Wiseman_, Jun 09 2024