A360402 -id:A360402 - cp's OEIS frontend

A360405 a(n) = A360393(A356133(n)).

2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, 124, 135, 142, 153, 163, 168, 180, 186, 195, 208, 217, 222, 231, 244, 249, 262, 271, 276, 285, 297, 307, 312, 324, 330, 339, 352, 361, 366, 375, 387, 394, 405, 414, 421, 432, 438, 447, 459, 466, 477

Offset: 1

Clark Kimberling, Apr 01 2023

This is the fourth of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) v o u, defined by (v o u)(n) = v(u(n));
(2) v' o u;
(3) v o u';
(4) v' o u.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively (and likewise for A360394-A360401).

			(1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
(2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
(3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
(4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405

Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360402, A360403, A360404.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v]; (* A360393 *)
Table[v[[u[[n]]]], {n, 1, zz}]     (* A360402 *)
Table[v1[[u[[n]]]], {n, 1, zz}]    (* A360403 *)
Table[v[[u1[[n]]]], {n, 1, zz}]    (* A360404 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]   (* A360405 *)

A360394 Intersection of A026430 and A360392.

Original entry on oeis.org

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 *)

A360398 a(n) = A026430(1 + A360392(n)).

Original entry on oeis.org

5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, 42, 44, 45, 48, 50, 52, 55, 57, 59, 61, 65, 66, 69, 70, 72, 75, 78, 80, 81, 84, 86, 88, 91, 93, 95, 98, 100, 102, 105, 107, 108, 111, 113, 116, 118, 120, 123, 125, 126, 129, 132, 134, 135, 138

Offset: 1

Clark Kimberling, Feb 10 2023

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360399, A286355, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];    (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A360401 a(n) = A356133(A360393(n)).

Original entry on oeis.org

2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, 124, 133, 146, 151, 164, 173, 178, 191, 197, 206, 218, 227, 233, 242, 253, 260, 272, 280, 287, 295, 308, 317, 322, 335, 341, 350, 362, 371, 377, 385, 398, 403, 415, 425, 430, 443, 449, 457, 470

Offset: 1

Clark Kimberling, Mar 11 2023

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286355, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A360399 a(n) = A026430(1 + A360393(n)).

Original entry on oeis.org

1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, 73, 77, 82, 87, 90, 96, 99, 103, 109, 114, 117, 121, 128, 130, 136, 141, 144, 149, 154, 159, 162, 168, 171, 175, 181, 186, 189, 194, 199, 203, 209, 213, 216, 222, 225, 230, 235, 239, 245, 249

Offset: 1

Clark Kimberling, Feb 10 2023

This is the second of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286355, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];    (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A360400 a(n) = A356133(A360392(n)).

Original entry on oeis.org

7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, 83, 85, 89, 94, 97, 104, 110, 112, 115, 122, 127, 131, 137, 140, 142, 148, 155, 157, 161, 166, 169, 176, 182, 184, 187, 193, 200, 202, 208, 211, 215, 220, 223, 229, 236, 238, 244, 247, 251, 257

Offset: 1

Clark Kimberling, Mar 11 2023

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A360403 a(n) = A360393(A026430(n)).

Original entry on oeis.org

1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, 66, 73, 76, 78, 85, 87, 94, 99, 103, 106, 112, 117, 121, 126, 129, 133, 139, 144, 148, 150, 157, 159, 166, 171, 175, 178, 184, 189, 193, 199, 202, 204, 210, 213, 220, 225, 229, 235, 238, 240, 246, 253

Offset: 1

Clark Kimberling, Mar 11 2023

This is the second of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) v o u, defined by (v o u)(n) = v(u(n));
(2) v' o u;
(3) v o u';
(4) v' o u.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively (and likewise for A360394-A360401).

			(1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
(2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
(3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
(4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360402, A360404, A360405.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Table[v[[u[[n]]]], {n, 1, zz}]    (* A360402 *)
Table[v1[[u[[n]]]], {n, 1, zz}]   (* A360403 *)
Table[v[[u1[[n]]]], {n, 1, zz}]   (* A360404 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]  (* A360405 *)

def A360393(n):
    if n < 3: return [0, 1, 2][n]
    return 3*n - 5 - (n-3).bit_count() % 2
def A026430(n): return n+(n-1>>1)+(n-1&1|(n.bit_count()&1^1))
def A360403(n): return A360393(A026430(n)) # Winston de Greef, Mar 24 2023

A360404 a(n) = A360392(A356133(n)).

Original entry on oeis.org

5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, 72, 75, 82, 86, 89, 95, 98, 102, 109, 113, 116, 120, 127, 130, 136, 140, 143, 147, 154, 158, 161, 167, 170, 174, 181, 185, 188, 192, 198, 201, 207, 212, 215, 221, 224, 228, 234, 237, 243, 248, 251

Offset: 1

Clark Kimberling, Apr 01 2023

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) v o u, defined by (v o u)(n) = v(u(n));
(2) v' o u;
(3) v o u';
(4) v' o u.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively (and likewise for A360394-A360401).

			(1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
(2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
(3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
(4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405

Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360402, A360403, A360405.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Table[v[[u[[n]]]], {n, 1, zz}]     (* A360402 *)
Table[v1[[u[[n]]]], {n, 1, zz}]    (* A360403 *)
Table[v[[u1[[n]]]], {n, 1, zz}]    (* A360404 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]   (* A360405 *)

cp's OEIS Frontend

A360405 a(n) = A360393(A356133(n)).

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A360394 Intersection of A026430 and A360392.

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A360398 a(n) = A026430(1 + A360392(n)).

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A360401 a(n) = A356133(A360393(n)).

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A360399 a(n) = A026430(1 + A360393(n)).

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A360400 a(n) = A356133(A360392(n)).

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A360403 a(n) = A360393(A026430(n)).

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Python

A360404 a(n) = A360392(A356133(n)).

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