A356104 a(n) = A000201(A022839(n)).

3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, 50, 53, 56, 61, 64, 67, 71, 74, 79, 82, 85, 88, 93, 97, 100, 103, 108, 111, 114, 118, 122, 126, 129, 132, 135, 140, 144, 147, 150, 155, 158, 161, 165, 169, 173, 176, 179, 184, 187, 190, 194, 197, 202, 205, 208

Offset: 1

Author

Clark Kimberling, Sep 08 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. For the reverse composites, v o u, v' o u, v o u', v' o u', see A356217 to A356220.

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 A356104, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.

Examples

			(1)  u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2)  u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3)  u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = A356106
(4)  u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107
		

Crossrefs

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356105, A356106, A356107, A351415 (intersections), A356217 (reverse composites).

Programs

Mathematica

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356106 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356107 *)