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Run-lengths (differing by 0) of antirun-lengths (differing by > 2) of odd primes.
12

%I #5 Jun 22 2024 18:05:35

%S 2,2,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,

%T 1,1,1,3,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1

%N Run-lengths (differing by 0) of antirun-lengths (differing by > 2) of odd primes.

%C Run-lengths of the version of A027833 with 1 prepended.

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

%e The antiruns of odd primes (differing by > 2) begin:

%e 3

%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 181 191

%e 193 197

%e 199 211 223 227

%e 229 233 239

%e 241 251 257 263 269

%e 271 277 281

%e with lengths:

%e 1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, ...

%e with runs:

%e 1 1

%e 2 2

%e 3 3

%e 4

%e 3

%e 6

%e 2

%e 5

%e 2

%e 6

%e 2 2

%e 4

%e 3

%e 5

%e 3

%e 4

%e with lengths a(n).

%t Length/@Split[Length/@Split[Select[Range[3,1000],PrimeQ],#2-#1>2&]//Most]//Most

%Y Run-lengths of A027833 (if we prepend 1), partial sums A029707.

%Y For runs we have A373819, run-lengths of A251092.

%Y Positions of first appearances are A373827, sorted A373826.

%Y A000040 lists the primes.

%Y A001223 gives differences of consecutive primes, run-lengths A333254, run-lengths of run-lengths A373821.

%Y A046933 counts composite numbers between primes.

%Y A065855 counts composite numbers up to n.

%Y A071148 gives partial sums of odd primes.

%Y For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.

%Y For composite runs: A005381, A054265, A068780, A373403, A373404.

%Y Cf. A005117, A006560, A006562, A037201, A038664, A049579, A073051, A076259, A122535, A176246.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 22 2024