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

2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, 131, 142, 148, 161, 169, 176, 187, 193, 202, 215, 223, 229, 238, 251, 257, 269, 278, 283, 292, 305, 313, 320, 331, 337, 346, 359, 367, 373, 382, 394, 400, 412, 421, 428, 439, 445, 454, 466, 472

Offset: 1

Author

Clark Kimberling, Jan 30 2023

Keywords

• nonn
• easy

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

(2) u o v';

(3) u' o v;

(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.

Examples

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135
		

Crossrefs

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352-A360134, A360136-A360139.

Programs

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[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)