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 1-7 of 7 results.

A356056 a(n) = A001951(A137803(n)).

Original entry on oeis.org

1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, 33, 36, 39, 42, 45, 48, 50, 53, 56, 59, 62, 63, 66, 69, 72, 74, 77, 80, 83, 86, 89, 91, 93, 96, 98, 101, 104, 107, 110, 113, 115, 118, 121, 124, 125, 128, 131, 134, 137, 140, 142, 145, 148, 151, 154, 156, 158, 161
Offset: 1

Views

Author

Clark Kimberling, Jul 26 2022

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) u' o v'.
Every positive integer is in exactly one of the four sequences.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356056, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*(1/2 + sqrt(2))), so that r = sqrt(2), s = 1/2 + sqrt(2), r' = 2 + sqrt(2), s' = (9 + 4*sqrt(2))/7.

Examples

			(1)  u o v = (1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, ...) = A356056
(2)  u o v' = (2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, ...) = A356057
(3)  u' o v = (3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, ...) = A356058
(4)  u' o v' = (6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, ...) = A356059

Crossrefs

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356057, A356058, A356059.

Programs

  • Mathematica
    z = 800;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]  (* A137804 *)
Table[u[[v[[n]]]], {n, 1, z/8}];   (* A356056 *)
Table[u[[v1[[n]]]], {n, 1, z/8}];  (* A356057 *)
Table[u1[[v[[n]]]], {n, 1, z/8}];  (* A356058 *)
Table[u1[[v1[[n]]]], {n, 1, z/8}]; (* A356059 *)

Formula

a(n) = A001951(A137803(n)).

A356057 a(n) = A001951(A137804(n)).

Original entry on oeis.org

2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, 38, 41, 43, 46, 49, 52, 55, 57, 60, 65, 67, 70, 73, 76, 79, 82, 84, 87, 90, 94, 97, 100, 103, 106, 108, 111, 114, 117, 120, 123, 127, 130, 132, 135, 138, 141, 144, 147, 149, 152, 155, 159, 162, 165, 168, 171, 173
Offset: 1

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

This is the second of four sequences that partition the positive integers. See A356056.

Examples

			(1)  u o v = (1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, ...) = A356056
(2)  u o v' = (2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, ...) = A356057
(3)  u' o v = (3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, ...) = A356058
(4)  u' o v' = (6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, ...) = A356059

Crossrefs

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356056, A356058, A356059.

Programs

  • Mathematica
    u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]  (* A137804 *)
Table[u[[v[[n]]]], {n, 1, z/8}];   (* A356056 *)
Table[u[[v1[[n]]]], {n, 1, z/8}];  (* A356057 *)
Table[u1[[v[[n]]]], {n, 1, z/8}];  (* A356058 *)
Table[u1[[v1[[n]]]], {n, 1, z/8}]; (* A356059 *)

Formula

a(n) = A001951(A137804(n)).

A356059 a(n) = A001952(A137804(n)).

Original entry on oeis.org

6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, 85, 92, 99, 105, 112, 119, 126, 133, 139, 146, 157, 163, 170, 177, 184, 191, 198, 204, 211, 218, 228, 235, 242, 249, 256, 262, 269, 276, 283, 290, 297, 307, 314, 320, 327, 334, 341, 348, 355, 361, 368, 375, 385, 392
Offset: 1

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

This is the fourth of four sequences that partition the positive integers. See A356056.

Examples

			(1)  u o v = (1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, ...) = A356056
(2)  u o v' = (2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, ...) = A356057
(3)  u' o v = (3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, ...) = A356058
(4)  u' o v' = (6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, ...) = A356059

Crossrefs

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356056, A356057, A356058.

Programs

  • Mathematica
    u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]  (* A137804 *)
Table[u[[v[[n]]]], {n, 1, z/8}];   (* A356056 *)
Table[u[[v1[[n]]]], {n, 1, z/8}];  (* A356057 *)
Table[u1[[v[[n]]]], {n, 1, z/8}];  (* A356058 *)
Table[u1[[v1[[n]]]], {n, 1, z/8}]; (* A356059 *)

Formula

a(n) = A001952(A137804(n)).

A356138 a(n) = A137803(A001951(n)).

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 17, 21, 22, 26, 28, 30, 34, 36, 40, 42, 45, 47, 49, 53, 55, 59, 61, 63, 66, 68, 72, 74, 78, 80, 82, 86, 88, 91, 93, 95, 99, 101, 105, 107, 109, 112, 114, 118, 120, 124, 126, 128, 132, 133, 137, 139, 141, 145, 147, 151, 153, 156, 158, 160
Offset: 1

Views

Author

Clark Kimberling, Aug 06 2022

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. For the reverse composites, u o v, u o v', u' o v, u' o v', see A356056 to A356059.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356138, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*(1/2 + sqrt(2))), so that r = sqrt(2), s = 1/2 + sqrt(2), r' = 2 + sqrt(2), s' = (9 + 4*sqrt(2))/7.

Examples

			(1)  v o u   = (1,  3,  7,  9, 13, 15, 17, 21, 22, 26, 28, 30, 34, ...) = A356138
(2)  v' o u  = (2,  4,  8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, ...) = A356139
(3)  v o u'  = (5, 11, 19, 24, 32, 38, 44, 51, 57, 65, 70, 76, 84, ...) = A356140
(4)  v' o u' = (6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, ...) = A356141

Crossrefs

Cf. A001951, A001952, A137804.
Cf. A356056, A356057, A356058, A356059, A356139, A356140, A356141.

Programs

  • Mathematica
    z = 800;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}];   (* A001951 *)
u1 = Complement[Range[Max[u]], u] ;    (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}];  (* A137803 *)
v1 = Complement[Range[Max[v]], v] ;     (* A137804 *)
Table[v[[u[[n]]]], {n, 1, z/8}]   (* A356138 *)
Table[v1[[u[[n]]]], {n, 1, z/8}]  (* A356139 *)
Table[v[[u1[[n]]]], {n, 1, z/8}]  (* A356140 *)
Table[v1[[u1[[n]]]], {n, 1, z/8}] (* A356141 *)

A356139 a(n) = A137804(A001951(n)).

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, 39, 43, 46, 50, 52, 54, 58, 60, 64, 67, 69, 73, 75, 79, 81, 85, 87, 90, 94, 96, 100, 102, 104, 108, 110, 115, 117, 119, 123, 125, 129, 131, 136, 138, 140, 144, 146, 150, 152, 154, 159, 161, 165, 167, 171, 173
Offset: 1

Views

Author

Clark Kimberling, Aug 06 2022

Keywords

Comments

This is the second of four sequences that partition the positive integers. See A356138.

Examples

			(1)  v o u   = (1,  3,  7,  9, 13, 15, 17, 21, 22, 26, 28, 30, 34, ...) = A356138
(2)  v' o u  = (2,  4,  8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, ...) = A356139
(3)  v o u'  = (5, 11, 19, 24, 32, 38, 44, 51, 57, 65, 70, 76, 84, ...) = A356140
(4)  v' o u' = (6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, ...) = A356141

Crossrefs

Cf. A001951, A001952, A137804.
Cf. A356056, A356057, A356058, A356059, A356138, A356140, A356141.

Programs

  • Mathematica
    z = 800;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}];   (*A001951*)
u1 = Complement[Range[Max[u]], u] ;    (*A001952*)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}];  (*A137803*)
v1 = Complement[Range[Max[v]], v] ;     (*A137804*)
Table[v[[u[[n]]]], {n, 1, z/8}]   (*A356138 *)
Table[v1[[u[[n]]]], {n, 1, z/8}]  (* A356139*)
Table[v[[u1[[n]]]], {n, 1, z/8}]  (* A356140 *)
Table[v1[[u1[[n]]]], {n, 1, z/8}] (* A356141 *)

A356140 a(n) = A137803(A001952(n)).

Original entry on oeis.org

5, 11, 19, 24, 32, 38, 44, 51, 57, 65, 70, 76, 84, 89, 97, 103, 111, 116, 122, 130, 135, 143, 149, 155, 162, 168, 176, 181, 189, 195, 200, 208, 214, 222, 227, 233, 241, 246, 254, 260, 266, 273, 279, 287, 292, 300, 306, 312, 319, 325, 333, 338, 344, 352, 357
Offset: 1

Views

Author

Clark Kimberling, Aug 06 2022

Keywords

Comments

This is the third of four sequences that partition the positive integers. See A356138.

Examples

			(1)  v o u   = (1,  3,  7,  9, 13, 15, 17, 21, 22, 26, 28, 30, 34, ...) = A356138
(2)  v' o u  = (2,  4,  8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, ...) = A356139
(3)  v o u'  = (5, 11, 19, 24, 32, 38, 44, 51, 57, 65, 70, 76, 84, ...) = A356140
(4)  v' o u' = (6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, ...) = A356141

Crossrefs

Cf. A001951, A001952, A137804.
Cf. A356056, A356057, A356058, A356059, A356138, A356139, A356141.

Programs

  • Mathematica
    z = 800;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}];   (*A001951*)
u1 = Complement[Range[Max[u]], u] ;    (*A001952*)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}];  (*A137803*)
v1 = Complement[Range[Max[v]], v] ;     (*A137804*)
Table[v[[u[[n]]]], {n, 1, z/8}]   (*A356138 *)
Table[v1[[u[[n]]]], {n, 1, z/8}]  (* A356139*)
Table[v[[u1[[n]]]], {n, 1, z/8}]  (* A356140 *)
Table[v1[[u1[[n]]]], {n, 1, z/8}] (* A356141 *)

A356141 a(n) = A137804(A001952(n)).

Original entry on oeis.org

6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, 98, 106, 113, 121, 127, 134, 142, 148, 157, 163, 169, 177, 184, 192, 198, 207, 213, 219, 228, 234, 242, 249, 255, 263, 270, 278, 284, 291, 299, 305, 314, 320, 328, 335, 341, 349, 355, 364, 370, 376, 385, 391
Offset: 1

Views

Author

Clark Kimberling, Aug 06 2022

Keywords

Comments

This is the fourth of four sequences that partition the positive integers. See A356138.

Examples

			(1)  v o u   = (1,  3,  7,  9, 13, 15, 17, 21, 22, 26, 28, 30, 34, ...) = A356138
(2)  v' o u  = (2,  4,  8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, ...) = A356139
(3)  v o u'  = (5, 11, 19, 24, 32, 38, 44, 51, 57, 65, 70, 76, 84, ...) = A356140
(4)  v' o u' = (6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, ...) = A356141

Crossrefs

Cf. A001951, A001952, A137804.
Cf. A356056, A356057, A356058, A356059, A356138, A356139, A356140.

Programs

  • Mathematica
    z = 800;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}];   (*A001951*)
u1 = Complement[Range[Max[u]], u] ;    (*A001952*)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}];  (*A137803*)
v1 = Complement[Range[Max[v]], v] ;     (*A137804*)
Table[v[[u[[n]]]], {n, 1, z/8}]   (*A356138 *)
Table[v1[[u[[n]]]], {n, 1, z/8}]  (* A356139*)
Table[v[[u1[[n]]]], {n, 1, z/8}]  (* A356140 *)
Table[v1[[u1[[n]]]], {n, 1, z/8}] (* A356141 *)
Showing 1-7 of 7 results.

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