A359352 -id:A359352 - cp's OEIS frontend

A359277 Intersection of A026430 and (1 + A285953).

6, 9, 10, 15, 16, 19, 24, 27, 28, 31, 36, 37, 42, 45, 46, 51, 52, 55, 60, 61, 66, 69, 70, 73, 78, 81, 82, 87, 88, 91, 96, 99, 100, 103, 108, 109, 114, 117, 118, 121, 126, 129, 130, 135, 136, 139, 144, 145, 150, 153, 154, 159, 160, 163, 168, 171, 172, 175

Offset: 1

Clark Kimberling, Jan 26 2023

This is the first of three sequences that partition the positive integers. Taking u = A026430 and v = 1 + A285953 (which is A285953 except for its initial 1), the three sequences are (1) u ^ v = intersection of u and v (in increasing order); (2) u ^ v'; and (3) u' ^ v. The limiting density of each of these is 1/3.

			(1)  u ^ v = (6, 9, 10, 15, 16, 19, 24, 27, 28, 31, 36, 37, ...) =    A359277
(2)  u ^ v' = (1, 3, 5, 8, 12, 14, 18, 21, 23, 26, 30, 33, 35, ...) =  A285953, except for the initial 1
(3)  u' ^ v = (2, 4, 7, 11, 13, 17, 20, 22, 25, 29, 32, 34, 38, ...) = A356133

Cf. A026530, A285954, A356133, A359352 to A360139) (results of compositions instead of intersections).

z = 200;
u = Accumulate[1 + ThueMorse /@ Range[0, z]]   (* A026430 *)
u1 = Complement[Range[Max[u]], u]  (* A356133 *)
v = u + 1
v1 = Complement[Range[Max[v]], v]
Intersection[u, v]    (* A359277 *)
Intersection[u, v1]   (* A285953 *)
Intersection[u1, v]   (* A356133 *)

A360139 a(n) = V(A356133(n)), where V(1) = 1 and V(k) = A285953(k+1) for k >= 2.

Original entry on oeis.org

3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, 125, 138, 143, 156, 165, 170, 183, 188, 198, 209, 219, 224, 234, 245, 252, 263, 273, 279, 288, 300, 309, 314, 327, 332, 342, 353, 363, 368, 378, 390, 395, 408, 416, 422, 435, 440, 450, 462, 467, 480

Offset: 1

Clark Kimberling, Feb 03 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.

			(1)  v o u = (2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, ...) = A360136
(2)  v' o u = (1, 5, 12, 14, 21, 23, 26, 33, 39, 41, 44, 50, 54, 59, 65, ...) = A360137
(3)  v o u' = (4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, ...) = A360138
(4)  v' o u' = (3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, ...) = A360139

Cf. A026530, A359352, A285953, A359277 (intersections instead of results of composition), A359352-A360138.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[v[[u[[n]]]], {n, 1, zz}]      (* A360136 *)
Table[v1[[u[[n]]]], {n, 1, zz}]     (* A360137 *)
Table[v[[u1[[n]]]], {n, 1, zz}]     (* A360138 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A360139 *)

A360135 a(n) = A356133(A285953(n+1)).

Original entry on oeis.org

2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, 131, 142, 148, 161, 169, 176, 187, 193, 202, 215, 223, 229, 238, 251, 257, 269, 278, 283, 292, 305, 313, 320, 331, 337, 346, 359, 367, 373, 382, 394, 400, 412, 421, 428, 439, 445, 454, 466, 472

Offset: 1

Clark Kimberling, Jan 30 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) 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, respectively.

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352-A360134, A360136-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)

A360136 a(n) = 1 + A026430(A026430(n)).

Original entry on oeis.org

2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, 37, 40, 42, 43, 46, 47, 51, 53, 55, 56, 60, 62, 64, 67, 69, 70, 73, 76, 78, 79, 82, 83, 87, 89, 91, 92, 96, 99, 100, 103, 104, 106, 109, 110, 114, 117, 118, 121, 122, 124, 127, 130, 131, 133, 136, 137

Offset: 1

Clark Kimberling, Feb 03 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) 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.

			(1)  v o u = (2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, ...) = A360136
(2)  v' o u = (1, 5, 12, 14, 21, 23, 26, 33, 39, 41, 44, 50, 54, 59, 65, ...) = A360137
(3)  v o u' = (4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, ...) = A360138
(4)  v' o u' = (3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, ...) = A360139

Cf. A026530, A359352, A285953, A359277 (intersections instead of results of composition), A359352-A360135, A360137-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];   (* A285953 *)
Table[v[[u[[n]]]], {n, 1, zz}]       (* A360136 *)
Table[v1[[u[[n]]]], {n, 1, zz}]      (* A360137 *)
Table[v[[u1[[n]]]], {n, 1, zz}]      (* A360138 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]     (* A360139 *)

def A360136(n): return 1+(m:=n+(n-1>>1)+(n-1&1|(n.bit_count()&1^1)))+(m-1>>1)+(m-1&1|(m.bit_count()&1^1)) # Chai Wah Wu, Mar 01 2023

A359353 a(n) = A026430(A285953(n+1)).

Original entry on oeis.org

1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, 75, 81, 86, 88, 95, 98, 102, 108, 113, 116, 120, 126, 129, 135, 139, 143, 147, 153, 158, 160, 167, 170, 174, 180, 185, 188, 192, 198, 201, 207, 212, 214, 221, 224, 228, 234, 237, 243, 248, 250

Offset: 1

Clark Kimberling, Jan 30 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, respectively.

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352, A360134-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)

A360134 a(n) = A356133(1 + A026430(n)).

Original entry on oeis.org

4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, 79, 83, 85, 92, 94, 101, 106, 110, 112, 119, 124, 127, 133, 137, 140, 146, 151, 155, 157, 164, 166, 173, 178, 182, 184, 191, 197, 200, 206, 208, 211, 218, 220, 227, 233, 236, 242, 244, 247, 253

Offset: 1

Clark Kimberling, Jan 30 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, respectively.

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352, A359353, A360135-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)

def A360134(n): return 3*(m:=n+1+(n-1>>1)+(n-1&1|(n.bit_count()&1^1)))-(2 if (m-1).bit_count()&1 else 1) # Chai Wah Wu, Mar 01 2023

A360137 a(n) = V(A026430(n)), where V(1) = 1 and V(k) = A285953(k+1) for k >= 2.

Original entry on oeis.org

1, 5, 12, 14, 21, 23, 26, 33, 39, 41, 44, 50, 54, 59, 65, 68, 75, 77, 80, 86, 90, 95, 102, 105, 107, 113, 120, 123, 128, 132, 134, 141, 147, 149, 152, 158, 162, 167, 174, 177, 179, 185, 192, 194, 201, 203, 207, 212, 216, 221, 228, 230, 237, 239, 243, 248

Offset: 1

Clark Kimberling, Feb 03 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.

			(1)  v o u = (2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, ...) = A360136
(2)  v' o u = (1, 5, 12, 14, 21, 23, 26, 33, 39, 41, 44, 50, 54, 59, 65, ...) = A360137
(3)  v o u' = (4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, ...) = A360138
(4)  v' o u' = (3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, ...) = A360139

Cf. A026530, A359352, A285953, A359277 (intersections instead of results of composition), A359352-A360136, A360138-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[v[[u[[n]]]], {n, 1, zz}]      (* A360136 *)
Table[v1[[u[[n]]]], {n, 1, zz}]     (* A360137 *)
Table[v[[u1[[n]]]], {n, 1, zz}]     (* A360138 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A360139 *)

A360138 a(n) = 1 + A026430(A356133(n)).

Original entry on oeis.org

4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, 71, 74, 81, 85, 88, 94, 97, 101, 108, 112, 115, 119, 126, 129, 135, 139, 142, 146, 153, 157, 160, 166, 169, 173, 180, 184, 187, 191, 197, 200, 206, 211, 214, 220, 223, 227, 233, 236, 242, 247, 250

Offset: 1

Clark Kimberling, Feb 03 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.

			(1)  v o u = (2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, ...) = A360136
(2)  v' o u = (1, 5, 12, 14, 21, 23, 26, 33, 39, 41, 44, 50, 54, 59, 65, ...) = A360137
(3)  v o u' = (4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, ...) = A360138
(4)  v' o u' = (3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, ...) = A360139

Cf. A026530, A359352, A285953, A359277 (intersections instead of results of composition), A359352-A360136, A360138-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[v[[u[[n]]]], {n, 1, zz}]      (* A360136 *)
Table[v1[[u[[n]]]], {n, 1, zz}]     (* A360137 *)
Table[v[[u1[[n]]]], {n, 1, zz}]     (* A360138 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A360139 *)

def A360138(n): return (m:=3*n-(2 if (n-1).bit_count()&1 else 1))+(m-1>>1)+(m-1&1|(m.bit_count()&1^1))+1 # Chai Wah Wu, Mar 01 2023

cp's OEIS Frontend

A359277 Intersection of A026430 and (1 + A285953).

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A360139 a(n) = V(A356133(n)), where V(1) = 1 and V(k) = A285953(k+1) for k >= 2.

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A360135 a(n) = A356133(A285953(n+1)).

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A360136 a(n) = 1 + A026430(A026430(n)).

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A359353 a(n) = A026430(A285953(n+1)).

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A360134 a(n) = A356133(1 + A026430(n)).

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A360137 a(n) = V(A026430(n)), where V(1) = 1 and V(k) = A285953(k+1) for k >= 2.

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A360138 a(n) = 1 + A026430(A356133(n)).

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