A360396 Intersection of A356133 and A360392.

7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, 67, 71, 74, 79, 83, 89, 92, 97, 101, 104, 110, 115, 119, 122, 127, 131, 137, 140, 146, 151, 155, 161, 164, 169, 173, 176, 182, 187, 191, 197, 200, 206, 211, 215, 218, 223, 227, 233, 236, 242, 247, 251, 257

Offset: 1

Author

Clark Kimberling, Feb 05 2023

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

For A360396, 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.

Examples

			(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
		

Crossrefs

Cf. A026430, A356133, A360392-A360395, A360397-A360405.

Programs

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 *)