A373402 Numbers k such that the k-th maximal antirun of prime numbers > 3 has length different from all prior maximal antiruns. Sorted list of positions of first appearances in A027833.

1, 2, 4, 6, 8, 10, 21, 24, 30, 35, 40, 41, 46, 50, 69, 82, 131, 140, 185, 192, 199, 210, 248, 251, 271, 277, 325, 406, 423, 458, 645, 748, 811, 815, 826, 831, 987, 1053, 1109, 1426, 1456, 1590, 1629, 1870, 1967, 2060, 2371, 2607, 2920, 2946, 3564, 3681, 4119

Offset: 1

Author

Gus Wiseman, Jun 10 2024

Keywords

• nonn

Comments

The unsorted version is A373401.

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 begin:
    5
    7  11
   19  23  29
   43  47  53  59
   73  79  83  89  97 101
  109 113 127 131 137
		

Links

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

Crossrefs

For squarefree runs we have the triple (1,3,5), firsts of A120992.

For prime runs we have the triple (1,2,3), firsts of A175632.

For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.

For squarefree antiruns: A373200, unsorted A373128, firsts of A373127.

For composite runs we have A373400, unsorted A073051.

The unsorted version is A373401, firsts of A027833.

For composite antiruns we have the triple (1,2,7), firsts of A373403.

A000040 lists the primes, differences A001223.

A002808 lists the composite numbers, differences A073783.

A046933 counts composite numbers between primes.

A065855 counts composite numbers up to n.

Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

Programs

Mathematica

t=Length/@Split[Select[Range[4,10000],PrimeQ],#1+2!=#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]