A356089 a(n) = A001951(A054406(n)).

2, 5, 9, 12, 15, 19, 22, 25, 29, 32, 36, 39, 42, 46, 49, 52, 56, 59, 62, 66, 69, 73, 76, 79, 83, 86, 89, 93, 96, 98, 103, 106, 110, 113, 115, 120, 123, 125, 130, 132, 137, 140, 142, 147, 149, 152, 156, 159, 162, 166, 169, 173, 176, 179, 183, 186, 189, 193

Offset

1,1

Comments

This is the second 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 A356089, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3), so that r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2.

Links

Table of n, a(n) for n=1..58.

Example

(1)  u o v   = (1,  4,  7,  8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088.
(2)  u o v'  = (2,  5,  9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089.
(3)  u' o v  = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090.
(4)  u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091.

Mathematica

z = 600; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];  (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356088 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356089 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356090 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356091 *)

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356088, A356090, A356091.

Sequence in context: A287409 A279171 A297465 * A108165 A063957 A184579

Adjacent sequences: A356086 A356087 A356088 * A356090 A356091 A356092

Keyword

nonn,easy

Author

Clark Kimberling, Aug 04 2022

Status

approved