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%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
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