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A373825 Position of first appearance of n in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

%I A373825 #5 Jun 22 2024 22:08:59
%S A373825 1,2,13,11,105,57,33,69,59,29,227,129,211,341,75,321,51,45,407,313,
%T A373825 459,301,767,1829,413,537,447,1113,1301,1411,1405,2865,1709,1429,3471,
%U A373825 709,2543,5231,1923,679,3301,2791,6555,5181,6345,11475,2491,10633
%N A373825 Position of first appearance of n in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.
%C A373825 Positions of first appearances in A373819.
%e A373825 The runs of odd primes differing by 2 begin:
%e A373825    3   5   7
%e A373825   11  13
%e A373825   17  19
%e A373825   23
%e A373825   29  31
%e A373825   37
%e A373825   41  43
%e A373825   47
%e A373825   53
%e A373825   59  61
%e A373825   67
%e A373825   71  73
%e A373825   79
%e A373825 with lengths:
%e A373825 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 A373825 which have runs beginning:
%e A373825   3
%e A373825   2 2
%e A373825   1
%e A373825   2
%e A373825   1
%e A373825   2
%e A373825   1 1
%e A373825   2
%e A373825   1
%e A373825   2
%e A373825   1 1 1 1
%e A373825   2 2
%e A373825   1 1 1
%e A373825 with lengths:
%e A373825 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3, ...
%e A373825 with positions of first appearances a(n).
%t A373825 t=Length/@Split[Length/@Split[Select[Range[3,10000], PrimeQ],#1+2==#2&]//Most]//Most;
%t A373825 spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
%t A373825 Table[Position[t,k][[1,1]],{k,spna[t]}]
%Y A373825 Firsts of A373819 (run-lengths of A251092).
%Y A373825 For antiruns we have A373827 (sorted A373826), firsts of A373820, run-lengths of A027833 (partial sums A029707, firsts A373401, sorted A373402).
%Y A373825 The sorted version is A373824.
%Y A373825 A000040 lists the primes.
%Y A373825 A001223 gives differences of consecutive primes (firsts A073051), run-lengths A333254 (firsts A335406), run-lengths of run-lengths A373821.
%Y A373825 A046933 counts composite numbers between primes.
%Y A373825 A065855 counts composite numbers up to n.
%Y A373825 A071148 gives partial sums of odd primes.
%Y A373825 For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
%Y A373825 For composite runs: A005381, A054265, A068780, A176246, A373403, A373404.
%Y A373825 Cf. A006560, A006562, A037201, A038664, A069010, A122535.
%K A373825 nonn
%O A373825 1,2
%A A373825 _Gus Wiseman_, Jun 21 2024