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A360400 a(n) = A356133(A360392(n)).

%I A360400 #4 Mar 24 2023 17:57:59
%S A360400 7,13,20,22,29,32,34,40,47,49,53,58,62,67,74,76,83,85,89,94,97,104,
%T A360400 110,112,115,122,127,131,137,140,142,148,155,157,161,166,169,176,182,
%U A360400 184,187,193,200,202,208,211,215,220,223,229,236,238,244,247,251,257
%N A360400 a(n) = A356133(A360392(n)).
%C A360400 This is the third 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 A360400 (1) u o v, defined by (u o v)(n)  = u(v(n));
%C A360400 (2) u o v';
%C A360400 (3) u' o v;
%C A360400 (4) v' o u'.
%C A360400 Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).
%e A360400 (1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
%e A360400 (2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
%e A360400 (3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
%e A360400 (4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401
%t A360400 z = 2000;
%t A360400 u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
%t A360400 u1 = Complement[Range[Max[u]], u];  (* A356133 *)
%t A360400 v = u + 2;  (* A360392 *)
%t A360400 v1 = Complement[Range[Max[v]], v];  (* A360393 *)
%t A360400 zz = 100;
%t A360400 Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
%t A360400 Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
%t A360400 Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
%t A360400 Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)
%Y A360400 Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286356, A360394 (intersections instead of results of composition), A360402-A360405.
%K A360400 nonn,easy
%O A360400 1,1
%A A360400 _Clark Kimberling_, Mar 11 2023