A360139 - OEIS

%I #6 Feb 04 2023 20:45:46

%S 3,8,18,30,35,48,57,63,72,84,93,98,111,116,125,138,143,156,165,170,

%T 183,188,198,209,219,224,234,245,252,263,273,279,288,300,309,314,327,

%U 332,342,353,363,368,378,390,395,408,416,422,435,440,450,462,467,480

%N a(n) = V(A356133(n)), where V(1) = 1 and V(k) = A285953(k+1) for k >= 2.

%C 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 (1) v o u, defined by (v o u)(n)  = v(u(n));

%C (2) v' o u;

%C (3) v o u';

%C (4) v' o u.

%C 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 (1)  v o u = (2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, ...) = A360136

%e (2)  v' o u = (1, 5, 12, 14, 21, 23, 26, 33, 39, 41, 44, 50, 54, 59, 65, ...) = A360137

%e (3)  v o u' = (4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, ...) = A360138

%e (4)  v' o u' = (3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, ...) = A360139

%t z = 2000; zz = 100;

%t u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)

%t u1 = Complement[Range[Max[u]], u];  (* A356133 *)

%t v = u + 1;  (* A285954 *)

%t v1 = Complement[Range[Max[v]], v];  (* A285953 *)

%t Table[v[[u[[n]]]], {n, 1, zz}]      (* A360136 *)

%t Table[v1[[u[[n]]]], {n, 1, zz}]     (* A360137 *)

%t Table[v[[u1[[n]]]], {n, 1, zz}]     (* A360138 *)

%t Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A360139 *)

%Y Cf. A026530, A359352, A285953, A359277 (intersections instead of results of composition), A359352-A360138.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Feb 03 2023