A360396 -id:A360396 - cp's OEIS frontend

A360394 Intersection of A026430 and A360392.

3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 57, 59, 61, 63, 65, 68, 70, 72, 75, 77, 80, 82, 84, 86, 88, 90, 93, 95, 98, 100, 102, 105, 107, 109, 111, 113, 116, 118, 120, 123, 125, 128, 130, 132, 134, 136, 138

Offset: 1

Clark Kimberling, Feb 05 2023

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. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively (and likewise for A360402-A360405).
For A360394, 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.

			(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

Cf. A026430, A356133, A360392, A360393, A356850-A360405.

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 *)

A360397 Intersection of A356133 and A360393.

Original entry on oeis.org

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

Clark Kimberling, Feb 10 2023

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.

			(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

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

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 *)

A360395 Intersection of A026430 and A360394.

Original entry on oeis.org

1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, 66, 69, 73, 78, 81, 87, 91, 96, 99, 103, 108, 114, 117, 121, 126, 129, 135, 139, 144, 150, 153, 159, 163, 168, 171, 175, 180, 186, 189, 195, 199, 204, 210, 213, 217, 222, 225, 231, 235, 240, 246, 249, 255

Offset: 1

Clark Kimberling, Feb 05 2023

This is the second 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 A360395, 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.

			(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

Cf. A026430, A356133, A360392-A360394, A360396-A360405.

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 *)

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A360394 Intersection of A026430 and A360392.

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A360397 Intersection of A356133 and A360393.

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A360395 Intersection of A026430 and A360394.

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