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A360405 a(n) = A360393(A356133(n)).

%I A360405 #7 Jun 21 2025 00:45:24
%S A360405 2,6,15,27,34,45,55,60,69,81,91,96,108,114,124,135,142,153,163,168,
%T A360405 180,186,195,208,217,222,231,244,249,262,271,276,285,297,307,312,324,
%U A360405 330,339,352,361,366,375,387,394,405,414,421,432,438,447,459,466,477
%N A360405 a(n) = A360393(A356133(n)).
%C A360405 This is the fourth of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
%C A360405 (1) v o u, defined by (v o u)(n) = v(u(n));
%C A360405 (2) v' o u;
%C A360405 (3) v o u';
%C A360405 (4) v' o u.
%C A360405 Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively (and likewise for A360394-A360401).
%e A360405 (1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
%e A360405 (2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
%e A360405 (3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
%e A360405 (4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405
%t A360405 z = 2000; zz = 100;
%t A360405 u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
%t A360405 u1 = Complement[Range[Max[u]], u];  (* A356133 *)
%t A360405 v = u + 2;  (* A360392 *)
%t A360405 v1 = Complement[Range[Max[v]], v]; (* A360393 *)
%t A360405 Table[v[[u[[n]]]], {n, 1, zz}]     (* A360402 *)
%t A360405 Table[v1[[u[[n]]]], {n, 1, zz}]    (* A360403 *)
%t A360405 Table[v[[u1[[n]]]], {n, 1, zz}]    (* A360404 *)
%t A360405 Table[v1[[u1[[n]]]], {n, 1, zz}]   (* A360405 *)
%Y A360405 Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360402, A360403, A360404.
%K A360405 nonn,easy
%O A360405 1,1
%A A360405 _Clark Kimberling_, Apr 01 2023