cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Clark Kimberling, Feb 03 2023

Keywords

Comments

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.

Examples

			(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

Crossrefs

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

Programs

  • Mathematica
    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 *)

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

Views

Author

Clark Kimberling, Feb 03 2023

Keywords

Comments

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.

Examples

			(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

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Python
    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

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

Views

Author

Clark Kimberling, Feb 03 2023

Keywords

Comments

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.

Examples

			(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

Crossrefs

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

Programs

  • Mathematica
    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 *)
Showing 1-3 of 3 results.

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