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

3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, 38, 41, 43, 44, 47, 48, 52, 54, 56, 57, 61, 63, 65, 68, 70, 71, 74, 77, 79, 80, 83, 84, 88, 90, 92, 93, 97, 100, 101, 104, 105, 107, 110, 111, 115, 118, 119, 122, 123, 125, 128, 131, 132, 134, 137

Offset

1,1

Comments

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:

(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 (and likewise for A360394-A360401).

Links

Winston de Greef, Table of n, a(n) for n = 1..10000

Example

(1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
(2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
(3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
(4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405

Mathematica

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Table[v[[u[[n]]]], {n, 1, zz}]    (* A360402 *)
Table[v1[[u[[n]]]], {n, 1, zz}    (* A360403 *)
Table[v[[u1[[n]]]], {n, 1, zz}]   (* A360404 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]  (* A360405 *)

Prog

(Python)
def A360392(n): return n+2+(n-1>>1)+(n-1&1|(n.bit_count()&1^1))
def A026430(n): return n+(n-1>>1)+(n-1&1|(n.bit_count()&1^1))
def A360402(n): return A360392(A026430(n)) # Winston de Greef, Mar 24 2023

Crossrefs

Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360403, A360404, A360405.

Sequence in context: A345168 A348612 A349799 * A188081 A188091 A190677

Adjacent sequences: A360399 A360400 A360401 * A360403 A360404 A360405

Keyword

nonn,easy

Author

Clark Kimberling, Mar 11 2023

Status

approved