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.

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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 *)

A137805 Self-inverse integer permutation induced by Beatty sequences for Sqrt(2)+1/2 and (4*Sqrt(2)+9)/7.

Original entry on oeis.org

2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 18, 17, 20, 19, 23, 25, 21, 27, 22, 29, 24, 31, 26, 33, 28, 35, 30, 37, 32, 39, 34, 41, 36, 43, 38, 46, 40, 48, 50, 42, 52, 44, 54, 45, 56, 47, 58, 49, 60, 51, 62, 53, 64, 55, 67, 57, 69, 59, 71, 73, 61, 75, 63, 77, 65, 79
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 11 2008

Keywords

Links

Formula

a(A137803(n)) = A137804(n) and a(A137804(n)) = A137803(n).

A356081 Numbers k such that A356052(k) = A356056(k).

Original entry on oeis.org

1, 3, 4, 6, 8, 14, 16, 17, 22, 25, 27, 28, 30, 38, 40, 67, 68, 74, 78, 82, 102, 104, 109, 110, 112, 126, 128, 132, 136, 140, 160, 164, 188
Offset: 1

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

Conjectures:
(1) This sequence is finite, with greatest term 188.
(2) The set {A356056(k) - A356052(k)}, for k >=1,
contains every integer >= -5.

Crossrefs

Cf. A001951, A137803, A137803, A137804, A356052, A356056.

Programs

  • Mathematica
    z = 1000000;
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 *)
t1 = Intersection[u, v];      (* A356052 *)
t2 = Table[u[[v[[n]]]], {n, 1, z/2}];  (* A356056 *)
length = Min[Length[t1], Length[t2]]
t = Take[t2, length] - Take[t1, length];
{Min[t], Max[t]}
Flatten[Position[t, 0]]
Previous Showing 11-15 of 15 results.

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