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A360135 a(n) = A356133(A285953(n+1)).

%I A360135 #8 Jan 31 2023 08:34:05
%S A360135 2,7,13,22,34,40,53,62,67,76,89,97,104,115,122,131,142,148,161,169,
%T A360135 176,187,193,202,215,223,229,238,251,257,269,278,283,292,305,313,320,
%U A360135 331,337,346,359,367,373,382,394,400,412,421,428,439,445,454,466,472
%N A360135 a(n) = A356133(A285953(n+1)).
%C A360135 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 A360135 (1) u o v, defined by (u o v)(n) = u(v(n));
%C A360135 (2) u o v';
%C A360135 (3) u' o v;
%C A360135 (4) v' o u'.
%C A360135 Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.
%e A360135 (1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
%e A360135 (2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
%e A360135 (3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
%e A360135 (4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135
%t A360135 z = 2000; zz = 100;
%t A360135 u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
%t A360135 u1 = Complement[Range[Max[u]], u];  (* A356133 *)
%t A360135 v = u + 1;  (* A285954 *)
%t A360135 v1 = Complement[Range[Max[v]], v];  (* A285953 *)
%t A360135 Table[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
%t A360135 Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
%t A360135 Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
%t A360135 Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)
%Y A360135 Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352-A360134, A360136-A360139.
%K A360135 nonn,easy
%O A360135 1,1
%A A360135 _Clark Kimberling_, Jan 30 2023