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A360402 a(n) = A360392(A026430(n)).

%I A360402 #18 Jun 20 2025 20:22:19
%S A360402 3,7,10,11,14,16,17,20,23,25,26,29,30,33,37,38,41,43,44,47,48,52,54,
%T A360402 56,57,61,63,65,68,70,71,74,77,79,80,83,84,88,90,92,93,97,100,101,104,
%U A360402 105,107,110,111,115,118,119,122,123,125,128,131,132,134,137
%N A360402 a(n) = A360392(A026430(n)).
%C A360402 This is the first 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 A360402 (1) v o u, defined by (v o u)(n) = v(u(n));
%C A360402 (2) v' o u;
%C A360402 (3) v o u';
%C A360402 (4) v' o u.
%C A360402 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).
%H A360402 Winston de Greef, <a href="/A360402/b360402.txt">Table of n, a(n) for n = 1..10000</a>
%e A360402 (1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
%e A360402 (2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
%e A360402 (3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
%e A360402 (4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405
%t A360402 z = 2000; zz = 100;
%t A360402 u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
%t A360402 u1 = Complement[Range[Max[u]], u];  (* A356133 *)
%t A360402 v = u + 2;  (* A360392 *)
%t A360402 v1 = Complement[Range[Max[v]], v];  (* A360393 *)
%t A360402 Table[v[[u[[n]]]], {n, 1, zz}]    (* A360402 *)
%t A360402 Table[v1[[u[[n]]]], {n, 1, zz}]   (* A360403 *)
%t A360402 Table[v[[u1[[n]]]], {n, 1, zz}]   (* A360404 *)
%t A360402 Table[v1[[u1[[n]]]], {n, 1, zz}]  (* A360405 *)
%o A360402 (Python)
%o A360402 def A360392(n): return n+2+(n-1>>1)+(n-1&1|(n.bit_count()&1^1))
%o A360402 def A026430(n): return n+(n-1>>1)+(n-1&1|(n.bit_count()&1^1))
%o A360402 def A360402(n): return A360392(A026430(n)) # _Winston de Greef_, Mar 24 2023
%Y A360402 Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360403, A360404, A360405.
%K A360402 nonn,easy
%O A360402 1,1
%A A360402 _Clark Kimberling_, Mar 11 2023