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 1-4 of 4 results.

A356088 a(n) = A001951(A022838(n)).

Original entry on oeis.org

1, 4, 7, 8, 11, 14, 16, 18, 21, 24, 26, 28, 31, 33, 35, 38, 41, 43, 45, 48, 50, 53, 55, 57, 60, 63, 65, 67, 70, 72, 74, 77, 80, 82, 84, 87, 90, 91, 94, 97, 100, 101, 104, 107, 108, 111, 114, 117, 118, 121, 124, 127, 128, 131, 134, 135, 138, 141, 144, 145
Offset: 1

Views

Author

Clark Kimberling, Aug 04 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 A356088, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3)), so that r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2.

Examples

			(1)  u o v   = (1,  4,  7,  8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088.
(2)  u o v'  = (2,  5,  9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089.
(3)  u' o v  = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090.
(4)  u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091.

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356089, A356090, A356091.

Programs

  • Mathematica
    z = 600; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];  (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[u[[v[[n]]]], {n, 1, zz}];    (* A356088 *)
Table[u[[v1[[n]]]], {n, 1, zz}];   (* A356089 *)
Table[u1[[v[[n]]]], {n, 1, zz}];   (* A356090 *)
Table[u1[[v1[[n]]]], {n, 1, zz}];  (* A356091 *)
  • Python
    from math import isqrt
def A356088(n): return isqrt(isqrt(3*n*n)**2<<1) # Chai Wah Wu, Aug 06 2022

A356089 a(n) = A001951(A054406(n)).

Original entry on oeis.org

2, 5, 9, 12, 15, 19, 22, 25, 29, 32, 36, 39, 42, 46, 49, 52, 56, 59, 62, 66, 69, 73, 76, 79, 83, 86, 89, 93, 96, 98, 103, 106, 110, 113, 115, 120, 123, 125, 130, 132, 137, 140, 142, 147, 149, 152, 156, 159, 162, 166, 169, 173, 176, 179, 183, 186, 189, 193
Offset: 1

Views

Author

Clark Kimberling, Aug 04 2022

Keywords

Comments

This is the second of four sequences that partition the positive integers. See A356088.

Examples

			(1)  u o v   = (1,  4,  7,  8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088.
(2)  u o v'  = (2,  5,  9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089.
(3)  u' o v  = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090.
(4)  u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091.

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356088, A356090, A356091.

Programs

  • Mathematica
    z = 600; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];  (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356088 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356089 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356090 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356091 *)

A356090 a(n) = A001952(A022838(n)).

Original entry on oeis.org

3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, 68, 75, 81, 85, 92, 99, 105, 109, 116, 122, 129, 133, 139, 146, 153, 157, 163, 170, 174, 180, 187, 194, 198, 204, 211, 218, 221, 228, 235, 242, 245, 252, 259, 262, 269, 276, 283, 286, 293, 300, 307, 310, 317, 324
Offset: 1

Views

Author

Clark Kimberling, Aug 04 2022

Keywords

Comments

This is the third of four sequences that partition the positive integers. See A356088.

Examples

			(1)  u o v   = (1,  4,  7,  8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088
(2)  u o v'  = (2,  5,  9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089
(3)  u' o v  = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090
(4)  u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356088, A356089, A356091.

Programs

  • Mathematica
    z = 600; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];  (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356088 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356089 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356090 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356091 *)

A356180 a(n) = A022838(A001951(n)).

Original entry on oeis.org

1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, 36, 38, 41, 43, 45, 48, 50, 53, 55, 57, 60, 62, 65, 67, 71, 72, 74, 77, 79, 83, 84, 86, 90, 91, 95, 96, 98, 102, 103, 107, 109, 112, 114, 116, 119, 121, 124, 126, 128, 131, 133, 136, 138, 142, 143, 145, 148
Offset: 1

Views

Author

Clark Kimberling, Aug 24 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 A356088 to A356091.
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 A356180, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3)), so r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2.

Examples

			(1)  v o u = (1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, ...) = A356180
(2)  v' o u = (2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, ...) = A356181
(3)  v o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, ...) = A356182
(4)  v' o u' = (7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, ...) = A356183

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356181, A356182, A356183.

Programs

  • Mathematica
    z = 800; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];       (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[v[[u[[n]]]], {n, 1, zz}]      (* A356180 *)
Table[v1[[u[[n]]]], {n, 1, zz}]     (* A356181 *)
Table[v[[u1[[n]]]], {n, 1, zz}]     (* A356182 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A356183 *)
  • Python
    from math import isqrt
def A356180(n): return isqrt(3*isqrt(n**2<<1)**2) # Chai Wah Wu, Sep 06 2022
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.