A360138 a(n) = 1 + A026430(A356133(n)).

4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, 71, 74, 81, 85, 88, 94, 97, 101, 108, 112, 115, 119, 126, 129, 135, 139, 142, 146, 153, 157, 160, 166, 169, 173, 180, 184, 187, 191, 197, 200, 206, 211, 214, 220, 223, 227, 233, 236, 242, 247, 250

Offset

1,1

Comments

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:

(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..56.

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 *)

Prog

(Python)
def A360138(n): return (m:=3*n-(2 if (n-1).bit_count()&1 else 1))+(m-1>>1)+(m-1&1|(m.bit_count()&1^1))+1 # Chai Wah Wu, Mar 01 2023

Crossrefs

Cf. A026530, A359352, A285953, A359277 (intersections instead of results of composition), A359352-A360136, A360138-A360139.

Sequence in context: A310764 A310765 A038835 * A310766 A097403 A126618

Adjacent sequences: A360135 A360136 A360137 * A360139 A360140 A360141

Keyword

nonn,easy

Author

Clark Kimberling, Feb 03 2023

Status

approved