A356219 Intersection of A001952 and A003151.

284, 287, 289, 292, 294, 296, 299, 301, 304, 306, 309, 311, 313, 316, 318, 321, 323, 325, 328, 330, 333, 335, 337, 340, 342, 345, 347, 350, 352, 354, 357, 359, 362, 364, 366, 369, 371, 374, 376, 379, 381, 383, 386, 388, 391, 393, 395, 398, 400

Offset

1,1

Comments

This is the third 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 complements, and assume that the following four sequences are infinite:

(1) u ^ v = intersection of u and v (in increasing order);

(2) u ^ v';

(3) u' ^ v;

(4) u' ^ 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 A356219, 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).

Links

Table of n, a(n) for n=1..49.

Example

(1)  u ^ v = (2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, ...) =  A003151
(2)  u ^ v' = (1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, ...) =  A001954
(3)  u' ^ v = (284, 287, 289, 292, 294, 296, 299, 301, 304, 306, ...) = A356219
(4)  u' ^ v' = (3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, ...) = A003152

Mathematica

z = 200;
r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
r1 = 1 + Sqrt[2]; v = Table[Floor[n*r1], {n, 1, z}]  (* A003151 *)
v1 = Take[Complement[Range[1000], v], z]  (* A003152 *)
t1 = Intersection[u, v]    (* A003151 *)
t2 = Intersection[u, v1]   (* A001954 *)
t3 = Intersection[u1, v]   (* A356219 *)
t4 = Intersection[u1, v1]  (* A001952 *)

Crossrefs

Cf. A001951, A001952, A003151, A003152, A001954, A184922 (results of compositions instead of intersections), A341239 (reversed compositions).

Sequence in context: A263670 A108826 A285890 * A061310 A333930 A259996

Adjacent sequences: A356216 A356217 A356218 * A356220 A356221 A356222

Keyword

nonn,easy

Author

Clark Kimberling, Nov 13 2022

Status

approved