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

Offset: 1

Author

Gus Wiseman, Jun 09 2024

Keywords

• nonn

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.

Examples

			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.
		

Links

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

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.

Programs

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]}]