A184922 a(n) = n + [rn/t] + [sn/t] + [un/t], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

2, 5, 9, 12, 16, 19, 22, 26, 29, 33, 36, 39, 43, 46, 50, 53, 57, 60, 63, 67, 70, 74, 77, 80, 84, 87, 91, 94, 98, 101, 104, 108, 111, 115, 118, 121, 125, 128, 132, 135, 138, 142, 145, 149, 152, 156, 159, 162, 166, 169, 173, 176, 179, 183, 186, 190, 193, 197, 200, 203, 207, 210, 214, 217, 220, 224, 227, 231, 234, 237, 241, 244, 248, 251, 255, 258, 261, 265, 268, 272, 275, 278, 282, 285, 289, 292, 296, 299, 302, 306, 309, 313, 316, 319, 323, 326, 330, 333, 337, 340, 343, 347, 350, 354, 357, 360, 364, 367, 371, 374, 377, 381, 384, 388, 391, 395, 398, 401, 405, 408

Offset: 1

Author

Clark Kimberling, Jan 26 2011

Keywords

• nonn

Comments

From Clark Kimberling, Jan 26 2011: (Start)

The sequences A184920-A184923 partition the positive integers:

A184920: 7, 15, 24, 31, 40, 48, 55, 64, ...

A184921: 3, 8, 13, 18, 23, 27, 32, 37, ...

A184922: 2, 5, 9, 12, 16, 19, 22, 26, 29, ...

A184923: 1, 4, 6, 10, 11, 14, 17, 20, 21, ...

Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, where h>=1, i>=1, j>=1, k>=1. The position of n*t in the joint ranking is n + [rn/t] + [sn/t] + [un/t], and likewise for the positions of n*s, n*s, and n*u.

(End)

Since [rn/t] = sqrt(2) - 1, [sn/t] = sqrt(2)/2, and [un/t] = 2 - sqrt(2)/2, we find using [-x] = -[x] - 1 for noninteger x, that a(n) = floor(n*(2+sqrt(2))) - 1 = A001952(n) - 1. - Michel Dekking, Feb 22 2018

From Clark Kimberling, Dec 27 2022: (Start)

This is the first of four sequences that partition the positive integers. Starting with a general overview, 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.

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 A184922, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).

(1) u o v = (2, 5, 9, 12, 16, 19, 22, 26, 29, 33, 36, 39, 43, ...) = A184922

(2) u o v' = (1, 4, 7, 8, 11, 14, 15, 18, 21, 24, 25, 28, 31, ...) = A188376

(3) u' o v = (6, 13, 23, 30, 40, 47, 54, 64, 71, 81, 88, 95, ...) = A359351

(4) u' o v' = (3, 10, 17, 20, 27, 34, 37, 44, 51, 58, 61, 68, ...) = A188396

For results of intersections instead of intersections, see A003151. For the reverse composites, v o u, v' o u, v o u', v' o u', see A341239.

(End)

Crossrefs

Cf. A184920, A184921, A184922, A184923, A188376, A188396, A359351.

Programs

Mathematica

z = 100; zz = 10;
u = Table[Floor[n Sqrt[2]], {n, 1, z}]
u1 = Complement[Range[Max[u]], u]
v = Table[Floor[n (1 + Sqrt[2])], {n, 1, z}]
v1 = Complement[Range[Max[v]], v]
Table[u[[v[[n]]]], {n, 1, zz}];   (* A184922 *)
Table[u[[v1[[n]]]], {n, 1, zz}];  (* A188376 *)
Table[u1[[v[[n]]]], {n, 1, zz}];  (* A359351 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]; (* A188396 *)

Formula

a(n) = floor(n*(2+sqrt(2))) - 1. - Michel Dekking, Feb 22 2018

Extensions

Name corrected by Michel Dekking, Feb 22 2018

Edited by Clark Kimberling, Dec 27 2022