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A354793 Hamming weight of A354783(n).

%I A354793 #25 Jul 28 2022 12:10:26
%S A354793 0,0,1,1,2,0,1,1,2,0,2,2,3,1,2,0,1,1,3,1,2,0,1,1,2,0,2,2,3,1,3,1,2,0,
%T A354793 1,1,2,0,2,2,3,1,3,1,2,0,1,1,2,0,2,2,3,1,2,0,1,1,3,1,2,0,2,2,3,1,3,1,
%U A354793 2,0,1,1,2,0,2,2,3,1,2,0,1,1,3,1,2,0,2,2,3,1,3,1,2,0,1,1,2,0,2,2,3,1,2
%N A354793 Hamming weight of A354783(n).
%C A354793 Conjecture: This sequence appears to have a simple structure. Encode it by making the following substitutions, in this order:
%C A354793 Replace the initial 28 terms 0011201120223120113120112022 by S (as usual, the start is irregular), then map:
%C A354793   3 1 3 -> 7
%C A354793   3 1 2 -> 6
%C A354793   1 2 0 1 1 2 0 2 2 -> 9
%C A354793   0 1 1 -> 2
%C A354793   0 2 2 -> 4
%C A354793 Then it appears that the encoded sequence is the concatenation of the following blocks:
%C A354793   S
%C A354793   79
%C A354793   79(6264)^1
%C A354793   79(6264)^1
%C A354793   79(6264)^3
%C A354793   79(6264)^3
%C A354793   79(6264)^15
%C A354793   79(6264)^15
%C A354793   79(6264)^31
%C A354793   79(6264)^31
%C A354793   79(6264)^63
%C A354793   79(6264)^63
%C A354793   79(6264)^127
%C A354793   79(6264)^127
%C A354793   ...
%C A354793 This is probably not the most efficient encoding, but I was happy to find any one that revealed the structure.
%C A354793 From _Michel Dekking_, Jul 23 2022: (Start)
%C A354793 The following is another way to present the conjecture above, which shows the close connection with sequence A355150.
%C A354793 Conjecture: It appears that this sequence is almost a periodic sequence, with period 12. Let x:=A354789.
%C A354793 If n > 28, n == 5 (mod 12) is not an element of x then (written as words)
%C A354793       a(n)a(n+1)...a(n+11) = 312011312022.
%C A354793 If n > 28, n == 5 (mod 12) is an element of x then
%C A354793       a(n)a(n+1)...a(n+11) = 313120112022.
%C A354793 (End)
%H A354793 N. J. A. Sloane, <a href="/A354793/b354793.txt">Table of n, a(n) for n = 1..4900</a>
%Y A354793 Cf. A354169, A354757, A354783, A355150.
%K A354793 nonn
%O A354793 1,5
%A A354793 _N. J. A. Sloane_, Jul 19 2022