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.

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

Views

Author

Clark Kimberling, Feb 10 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) 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).

Examples

			(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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, Mar 11 2023

Keywords

Comments

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).

Examples

			(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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, Feb 10 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) 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).

Examples

			(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

Crossrefs

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

Programs

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

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