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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.
10

%I #8 Jun 10 2024 23:35:51

%S 1,2,4,6,8,10,21,24,30,35,40,41,46,50,69,82,131,140,185,192,199,210,

%T 248,251,271,277,325,406,423,458,645,748,811,815,826,831,987,1053,

%U 1109,1426,1456,1590,1629,1870,1967,2060,2371,2607,2920,2946,3564,3681,4119

%N 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.

%C The unsorted version is A373401.

%C For this sequence, we define an antirun to be an interval of positions at which consecutive primes differ by at least 3.

%H Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.

%e The maximal antiruns of prime numbers > 3 begin:

%e 5

%e 7 11

%e 13 17

%e 19 23 29

%e 31 37 41

%e 43 47 53 59

%e 61 67 71

%e 73 79 83 89 97 101

%e 103 107

%e 109 113 127 131 137

%e 139 149

%e 151 157 163 167 173 179

%e The a(n)-th rows begin:

%e 5

%e 7 11

%e 19 23 29

%e 43 47 53 59

%e 73 79 83 89 97 101

%e 109 113 127 131 137

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

%t Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

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

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

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

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

%Y For composite runs we have A373400, unsorted A073051.

%Y The unsorted version is A373401, firsts of A027833.

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

%Y A000040 lists the primes, differences A001223.

%Y A002808 lists the composite numbers, differences A073783.

%Y A046933 counts composite numbers between primes.

%Y A065855 counts composite numbers up to n.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 10 2024