A356101 Intersection of A000201 and A022839.

1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, 43, 45, 48, 50, 56, 59, 61, 63, 66, 72, 74, 77, 79, 85, 88, 90, 92, 95, 97, 101, 103, 106, 108, 110, 113, 119, 121, 124, 126, 132, 135, 137, 139, 142, 144, 148, 150, 153, 155, 161, 166, 168, 171, 173, 177, 179

Offset: 1

Author

Clark Kimberling, Sep 04 2022

Keywords

• nonn
• easy

Comments

This is the second of four sequences that partition the positive integers. See A351415.

Examples

			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 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, 37, ...) =  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, 54, ...) = A356103
		

Crossrefs

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, 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 *)