A373824 - OEIS

%I #6 Jun 22 2024 22:09:03

%S 1,2,11,13,29,33,45,51,57,59,69,75,105,129,211,227,301,313,321,341,

%T 407,413,447,459,537,679,709,767,1113,1301,1405,1411,1429,1439,1709,

%U 1829,1923,2491,2543,2791,2865,3301,3471,3641,4199,4611,5181,5231,6345,6555

%N Sorted positions of first appearances in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

%C Sorted positions of first appearances in A373819.

%e The runs of odd primes differing by 2 begin:

%e    3   5   7

%e   11  13

%e   17  19

%e   23

%e   29  31

%e   37

%e   41  43

%e   47

%e   53

%e   59  61

%e   67

%e   71  73

%e   79

%e with lengths:

%e 3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, ...

%e which have runs beginning:

%e   3

%e   2 2

%e   1

%e   2

%e   1

%e   2

%e   1 1

%e   2

%e   1

%e   2

%e   1 1 1 1

%e   2 2

%e   1 1 1

%e with lengths:

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

%e with sorted positions of first appearances a(n).

%t t=Length/@Split[Length/@Split[Select[Range[3,10000],PrimeQ],#1+2==#2&]];

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

%Y Sorted firsts of A373819 (run-lengths of A251092).

%Y The unsorted version is A373825.

%Y For antiruns we have A373826, unsorted A373827.

%Y A000040 lists the primes.

%Y A001223 gives differences of consecutive primes (firsts A073051), run-lengths A333254 (firsts A335406), 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 A373820 gives run-lengths of antirun-lengths, run-lengths of A027833.

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

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

%Y Cf. A006560, A006562, A029707, A037201, A038664, A069010, A122535, A176246, A373401, A373402.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 21 2024