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A385892 In the sequence of run lengths of binary indices of each positive integer (A245563), remove all duplicate rows after the first and take the last term of each remaining row.

%I A385892 #6 Jul 20 2025 15:01:46
%S A385892 1,2,1,3,1,2,4,1,1,2,3,5,1,1,1,2,2,3,4,6,1,1,1,1,1,2,2,2,3,3,4,5,7,1,
%T A385892 1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5,6,8,1,1,1,1,1,1,1,1,1,1,1,1,1,2,
%U A385892 2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,6,7
%N A385892 In the sequence of run lengths of binary indices of each positive integer (A245563), remove all duplicate rows after the first and take the last term of each remaining row.
%C A385892 A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
%e A385892 The binary indices of 53 are {1,3,5,6}, with maximal runs ((1),(3),(5,6)), with lengths (1,1,2), which is the 16th row of A385817, so a(16) = 2.
%t A385892 Last/@DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2==#1+1&],{n,100}]]
%Y A385892 In the following references, "before" is short for "before removing duplicate rows".
%Y A385892 Positions of firsts appearances appear to be A000071.
%Y A385892 Without the removals we have A090996.
%Y A385892 For sum instead of last we have A200648, before A000120.
%Y A385892 For length instead of last we have A200650+1, before A069010 = A037800+1.
%Y A385892 Last term of row n of A385817 (ranks A385818, before A385889), first A083368.
%Y A385892 A245563 gives run lengths of binary indices, see A245562, A246029, A328592.
%Y A385892 A384877 gives anti-run lengths of binary indices, A385816.
%Y A385892 Cf. A001629, A044813, A048793, A164707, A200649, A384878, A384890, A385886.
%K A385892 nonn
%O A385892 1,2
%A A385892 _Gus Wiseman_, Jul 18 2025