A351415 Intersection of Beatty sequences for (1+sqrt(5))/2 and sqrt(5).

4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, 46, 51, 53, 55, 58, 64, 67, 69, 71, 76, 80, 82, 84, 87, 93, 98, 100, 105, 111, 114, 116, 118, 122, 127, 129, 131, 134, 140, 145, 147, 152, 156, 158, 160, 163, 165, 169, 174, 176, 181, 187, 190, 192, 194, 199

Offset: 1

Author

Clark Kimberling, Feb 10 2022

Keywords

• nonn

Comments

Conjecture: every term of the difference sequence is in {2,3,4,5,6}, and each occurs infinitely many times.

From Clark Kimberling, Jul 29 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 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. For A351415, 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 that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.

(1) u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) = A351415

(2) u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, ...) = A356101

(3) u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102

(4) u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, ...) = A356103

(End)

Examples

			The two Beatty sequences are (1,3,4,6,8,9,11,12,14,...) and (2,4,6,8,11,13,15,17,...), with common terms forming the sequence (4,6,8,11,...).
		

Crossrefs

Cf. A000201, A022839, A296184, A346308, A347469, A347793.

Cf. A001950, A108598, A356101, A356102, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

Programs

Mathematica

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]    (* A351415 *)
Intersection[u, v1]   (* A356101 *)
Intersection[u1, v]   (* A356102 *)
Intersection[u1, v1]  (* A356103 *)