A356056 a(n) = A001951(A137803(n)).

1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, 33, 36, 39, 42, 45, 48, 50, 53, 56, 59, 62, 63, 66, 69, 72, 74, 77, 80, 83, 86, 89, 91, 93, 96, 98, 101, 104, 107, 110, 113, 115, 118, 121, 124, 125, 128, 131, 134, 137, 140, 142, 145, 148, 151, 154, 156, 158, 161

Offset: 1

Author

Clark Kimberling, Jul 26 2022

Keywords

• nonn
• easy

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

(2) u o v';

(3) u' o v;

(4) u' o v'.

Every positive integer is in exactly one of the four sequences.

Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and

1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.

For A356056, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*(1/2 + sqrt(2))), so that r = sqrt(2), s = 1/2 + sqrt(2), r' = 2 + sqrt(2), s' = (9 + 4*sqrt(2))/7.

Examples

			(1)  u o v = (1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, ...) = A356056
(2)  u o v' = (2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, ...) = A356057
(3)  u' o v = (3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, ...) = A356058
(4)  u' o v' = (6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, ...) = A356059
		

Crossrefs

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356057, A356058, A356059.

Programs

Mathematica

z = 800;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]  (* A137804 *)
Table[u[[v[[n]]]], {n, 1, z/8}];   (* A356056 *)
Table[u[[v1[[n]]]], {n, 1, z/8}];  (* A356057 *)
Table[u1[[v[[n]]]], {n, 1, z/8}];  (* A356058 *)
Table[u1[[v1[[n]]]], {n, 1, z/8}]; (* A356059 *)

Formula

a(n) = A001951(A137803(n)).