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

Author

Clark Kimberling, Nov 13 2022

Keywords

• nonn
• easy

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).

Examples

			(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.
		

Crossrefs

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

Programs

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 *)