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.

A356104 a(n) = A000201(A022839(n)).

Original entry on oeis.org

3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, 50, 53, 56, 61, 64, 67, 71, 74, 79, 82, 85, 88, 93, 97, 100, 103, 108, 111, 114, 118, 122, 126, 129, 132, 135, 140, 144, 147, 150, 155, 158, 161, 165, 169, 173, 176, 179, 184, 187, 190, 194, 197, 202, 205, 208
Offset: 1

Views

Author

Clark Kimberling, Sep 08 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. For the reverse composites, v o u, v' o u, v o u', v' o u', see A356217 to A356220.
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 A356104, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.

Examples

			(1)  u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2)  u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3)  u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = A356106
(4)  u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107

Crossrefs

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356105, A356106, A356107, A351415 (intersections), A356217 (reverse composites).

Programs

  • Mathematica
    z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356106 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356107 *)

A356107 a(n) = A001950(A108598(n)).

Original entry on oeis.org

2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, 60, 65, 70, 73, 78, 83, 89, 94, 96, 102, 107, 112, 117, 123, 125, 130, 136, 141, 146, 149, 154, 159, 164, 170, 172, 178, 183, 188, 193, 196, 201, 206, 212, 217, 222, 225, 230, 235, 240, 246, 248, 253, 259, 264
Offset: 1

Views

Author

Clark Kimberling, Oct 02 2022

Keywords

Comments

This is the fourth of four sequences that partition the positive integers. See A356104.

Examples

			(1)  u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2)  u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3)  u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = A356106
(4)  u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107

Crossrefs

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356104, A356105, A356106, A351415 (intersections), A356217 (reverse composites).

Programs

  • Mathematica
    z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356106 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356107 *)

A356105 a(n) = A000201(A108598(n)).

Original entry on oeis.org

1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, 40, 43, 45, 48, 51, 55, 58, 59, 63, 66, 69, 72, 76, 77, 80, 84, 87, 90, 92, 95, 98, 101, 105, 106, 110, 113, 116, 119, 121, 124, 127, 131, 134, 137, 139, 142, 145, 148, 152, 153, 156, 160, 163, 166, 168, 171
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2022

Keywords

Comments

This is the second of four sequences that partition the positive integers. See A356104.

Examples

			(1)  u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2)  u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3)  u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = A356106
(4)  u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107

Crossrefs

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356104, A356106, A356107, A351415 (intersections), A356217 (reverse composites).

Programs

  • Mathematica
    z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356106 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356107 *)
Showing 1-3 of 3 results.

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