A360139 a(n) = V(A356133(n)), where V(1) = 1 and V(k) = A285953(k+1) for k >= 2.

3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, 125, 138, 143, 156, 165, 170, 183, 188, 198, 209, 219, 224, 234, 245, 252, 263, 273, 279, 288, 300, 309, 314, 327, 332, 342, 353, 363, 368, 378, 390, 395, 408, 416, 422, 435, 440, 450, 462, 467, 480

Offset

1,1

Comments

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:

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

(2) v' o u;

(3) v o u';

(4) v' o u.

Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

Links

Table of n, a(n) for n=1..54.

Example

(1)  v o u = (2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, ...) = A360136
(2)  v' o u = (1, 5, 12, 14, 21, 23, 26, 33, 39, 41, 44, 50, 54, 59, 65, ...) = A360137
(3)  v o u' = (4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, ...) = A360138
(4)  v' o u' = (3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, ...) = A360139

Mathematica

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[v[[u[[n]]]], {n, 1, zz}]      (* A360136 *)
Table[v1[[u[[n]]]], {n, 1, zz}]     (* A360137 *)
Table[v[[u1[[n]]]], {n, 1, zz}]     (* A360138 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A360139 *)

Crossrefs

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

Sequence in context: A328054 A171701 A288249 * A004210 A247022 A119881

Adjacent sequences: A360136 A360137 A360138 * A360140 A360141 A360142

Keyword

nonn,easy

Author

Clark Kimberling, Feb 03 2023

Status

approved