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 1-10 of 17 results. Next

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)).

A356052 Intersection of A001951 and A137803.

Original entry on oeis.org

1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, 32, 36, 38, 42, 45, 49, 53, 55, 57, 59, 63, 65, 66, 70, 72, 74, 76, 80, 82, 84, 86, 89, 91, 93, 97, 101, 103, 107, 111, 114, 118, 120, 124, 128, 130, 132, 135, 137, 141, 145, 147, 149, 151, 155, 156, 158, 162, 164
Offset: 1

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

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.
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 A356052, 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 ^ v = (1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, ...) = A356052
(2)  u ^ v' = (2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, ...) = A356053
(3)  u' ^ v = (3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, ...) = A356054
(4)  u' ^ v' = (6, 10, 20, 23, 27, 37, 54, 58, 64, 71, 75, ...) = A356055

Crossrefs

Cf. A001951, A001952, A136803, A137804, A356053, A356054, A356055, A356056 (composites instead of intersections), A356081.

Programs

  • Mathematica
    z = 250;
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 *)
Intersection[u, v]    (* A356052 *)
Intersection[u, v1]   (* A356053 *)
Intersection[u1, v]   (* A356054 *)
Intersection[u1, v1]  (* A356055 *)

A356058 a(n) = A001952(A137803(n)).

Original entry on oeis.org

3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, 75, 81, 88, 95, 102, 109, 116, 122, 129, 136, 143, 150, 153, 160, 167, 174, 180, 187, 194, 201, 208, 215, 221, 225, 232, 238, 245, 252, 259, 266, 273, 279, 286, 293, 300, 303, 310, 317, 324, 331, 338, 344, 351, 358
Offset: 1

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

This is the third 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, ...) = A356056
(2)  u o v'  = (2,  5,  8, 11, 14, 16, 19, 22, 25, 28, 32, ...) = 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 the results of composition), A356056, A356057, 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) = A001952(A137803(n)).

A356054 Intersection of A001952 and A137803.

Original entry on oeis.org

3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, 78, 88, 95, 99, 105, 109, 112, 116, 122, 126, 133, 139, 143, 153, 160, 170, 174, 187, 191, 204, 208, 218, 225, 235, 245, 252, 256, 262, 266, 269, 273, 279, 283, 290, 300, 310, 317, 327, 331, 334, 338, 344, 348
Offset: 1

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

This is the third of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A356052.

Examples

			(1)  u ^ v = (1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, ...) =  A356052
(2)  u ^ v' = (2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, ...) =  A356053
(3)  u' ^ v = (3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, ...) = A356054
(4)  u' ^ v' = (6, 10, 20, 23, 27, 37, 54, 58, 64, 71, 75, ...) = A356055

Crossrefs

Cf. A001951, A001952, A136803, A137804, A356052, A356054, A356055, A356056 (composites instead of intersections).

Programs

  • Mathematica
    z = 250;
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 *)
Intersection[u, v]    (* A356052 *)
Intersection[u, v1]   (* A356053 *)
Intersection[u1, v]   (* A356054 *)
Intersection[u1, v1]  (* A356055 *)

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 *)

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 *)

A137804 a(n) = floor(n*(4*sqrt(2)+9)/7).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 134, 136
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 11 2008

Keywords

Comments

a(n) = A059534(n) for n <= 31;
Beatty sequence for (4*Sqrt(2)+9)/7; complement of A137803;
a(n) = A137805(A137803(n)) and A137805(a(n)) = A137803(n).

Links

Crossrefs

Cf. A059534, A137803, A137805

Programs

  • Magma
    [Floor(n*(4*Sqrt(2)+9)/7): n in [1..100]]; // G. C. Greubel, Mar 28 2018
  • Maple
    [seq(floor(n*(4*(sqrt(2))+9)/7),n=1..70)]; # Muniru A Asiru, Mar 29 2018
  • Mathematica
    Table[Floor[n*(4*Sqrt[2]+9)/7], {n,1,100}] (* G. C. Greubel, Mar 28 2018 *)
  • PARI
    for(n=1,100, print1(floor(n*(4*sqrt(2)+9)/7), ", ")) \\ G. C. Greubel, Mar 28 2018

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)).

A356053 Intersection of A001951 and A137804.

Original entry on oeis.org

2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, 39, 41, 43, 46, 48, 50, 52, 56, 60, 62, 67, 69, 73, 77, 79, 83, 87, 90, 94, 96, 98, 100, 104, 106, 108, 110, 113, 115, 117, 121, 123, 125, 127, 131, 134, 138, 140, 142, 144, 148, 152, 154, 159, 161, 165, 169, 171
Offset: 1

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

This is the second of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A356052.

Examples

			(1)  u ^ v = (1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, ...) =  A356052
(2)  u ^ v' = (2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, ...) =  A356053
(3)  u' ^ v = (3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, ...) = A356054
(4)  u' ^ v' = (6, 10, 20, 23, 27, 37, 54, 58, 64, 71, 75, ...) = A356055

Crossrefs

Cf. A001951, A001952, A136803, A137804, A356052, A356054, A356055, A356056 (composites instead of intersections).

Programs

  • Mathematica
    z = 250;
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 *)
Intersection[u, v]    (* A356052 *)
Intersection[u, v1]   (* A356053 *)
Intersection[u1, v]   (* A356054 *)
Intersection[u1, v1]  (* A356055 *)
Showing 1-10 of 17 results. Next

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