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

1,2

Comments

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

Links

Table of n, a(n) for n=1..59.

Example

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

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 *)

Crossrefs

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356102, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

Sequence in context: A310318 A310319 A073105 * A073536 A077468 A089425

Adjacent sequences: A356098 A356099 A356100 * A356102 A356103 A356104

Keyword

nonn,easy

Author

Clark Kimberling, Sep 04 2022

Status

approved