A360397 Intersection of A356133 and A360393.

2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, 112, 124, 133, 142, 148, 157, 166, 178, 184, 193, 202, 208, 220, 229, 238, 244, 253, 262, 274, 280, 292, 301, 310, 322, 328, 337, 346, 352, 364, 373, 382, 394, 400, 412, 421, 430, 436, 445, 454, 466, 472

Offset

1,1

Comments

This is the fourth 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. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively.

For A360397, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:

u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430;

u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u;

v = u + 1 = A285954, except its initial 1;

v' = complement of v.

Links

Table of n, a(n) for n=1..54.

Example

(1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
(2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
(3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397

Mathematica

z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v];    (* A360393 *)
Intersection[u, v]    (* A360394 *)
Intersection[u, v1]   (* A360395 *)
Intersection[u1, v]   (* A360396 *)
Intersection[u1, v1]  (* A360397 *)

Crossrefs

Cf. A026430, A356133, A360392-A360396, A360398-A360405.

Sequence in context: A356288 A082015 A362263 * A018263 A018587 A163986

Adjacent sequences: A360394 A360395 A360396 * A360398 A360399 A360400

Keyword

nonn,easy

Author

Clark Kimberling, Feb 10 2023

Status

approved