A356104 -id:A356104 - cp's OEIS frontend

A190509 a(n) = n + [nr/s] + [nt/s] + [nu/s] where r=golden ratio, s=r^2, t=r^3, u=r^4, and [] represents the floor function.

4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 69, 76, 80, 87, 91, 98, 105, 109, 116, 120, 127, 134, 138, 145, 152, 156, 163, 167, 174, 181, 185, 192, 199, 203, 210, 214, 221, 228, 232, 239, 243, 250, 257, 261, 268, 275, 279, 286, 290, 297, 304, 308, 315, 319, 326, 333, 337, 344, 351, 355, 362, 366, 373, 380, 384, 391, 398, 402, 409

Offset: 1

Clark Kimberling, May 11 2011

nonn

See A190508.
From Clark Kimberling, Nov 13 2022: (Start)
This is the third 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) u o v';
(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 A356104 to A356107.
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 this sequence, 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 that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217
(2)  v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218
(3)  v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = this sequence
(4)  v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Weiru Chen and Jared Krandel, Interpolating Classical Partitions of the Set of Positive Integers, arXiv:1810.11938 [math.NT], 2018. See sequence D1 p. 4. Also in The Ramanujan Journal, (2020).

Cf. A054770, A190508, A190511.

Cf. A000201, A001622, A003623, A340429.

[3*Floor(n*(Sqrt(5)+1)/2) + n: n in [1..80]]; // Vincenzo Librandi, Nov 01 2018

r:=(1+sqrt(5))/2: s:=r^2: t:=r^3: u:=r^4: a:=n->n+floor(n*r/s)+floor(n*t/s)+floor(n*u/s):  seq(a(n),n=1..70); # Muniru A Asiru, Nov 01 2018

(See A190508.)
Table[3 Floor[n (Sqrt[5] + 1) / 2] + n, {n, 1, 100}] (* Vincenzo Librandi, Nov 01 2018 *)

a(n) = 3*floor(n*(sqrt(5)+1)/2) + n; \\ Michel Marcus, Sep 10 2017; after Michel Dekking's formula

from math import isqrt
def A190509(n): return n+((m:=n+isqrt(5*n**2))&-2)+(m>>1) # Chai Wah Wu, Aug 10 2022

A190508: a(n) = n + [nr] + [nr^2] + [nr^3];
A190509: b(n) = [n/r] + n + [nr] + [nr^2];
A054770: c(n) = [n/r^2] + [n/r] + n + [nr];
A190511: d(n) = [n/r^3] + [n/r^2] + [n/r] + n.
a(n) = 3*A000201(n)+n, since r/s = 1/r = r-1, and u/s = r^2 = r+1. - Michel Dekking, Sep 06 2017
a(n) = A000201(n) + A003623(n). - Primoz Pirnat, Jan 08 2021

A351415 Intersection of Beatty sequences for (1+sqrt(5))/2 and sqrt(5).

Original entry on oeis.org

4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, 46, 51, 53, 55, 58, 64, 67, 69, 71, 76, 80, 82, 84, 87, 93, 98, 100, 105, 111, 114, 116, 118, 122, 127, 129, 131, 134, 140, 145, 147, 152, 156, 158, 160, 163, 165, 169, 174, 176, 181, 187, 190, 192, 194, 199

Offset: 1

Clark Kimberling, Feb 10 2022

nonn

Conjecture: every term of the difference sequence is in {2,3,4,5,6}, and each occurs infinitely many times.
From Clark Kimberling, Jul 29 2022: (Start)
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. For A351415, 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 that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) = A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, ...) =    A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) =  A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, ...) =  A356103
(End)

			The two Beatty sequences are (1,3,4,6,8,9,11,12,14,...) and (2,4,6,8,11,13,15,17,...), with common terms forming the sequence (4,6,8,11,...).

Cf. A000201, A022839, A296184, A346308, A347469, A347793.

Cf. A001950, A108598, A356101, A356102, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]    (* A351415 *)
Intersection[u, v1]   (* A356101 *)
Intersection[u1, v]   (* A356102 *)
Intersection[u1, v1]  (* A356103 *)

A356217 a(n) = A022839(A000201(n)).

Original entry on oeis.org

2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, 46, 49, 53, 55, 60, 64, 67, 71, 73, 78, 82, 84, 89, 93, 96, 100, 102, 107, 111, 114, 118, 122, 125, 129, 131, 136, 140, 143, 147, 149, 154, 158, 160, 165, 169, 172, 176, 178, 183, 187, 190, 194, 196, 201, 205, 207

Offset: 1

Clark Kimberling, Oct 02 2022

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 A356104 to A356107.
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 A356217 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 that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.

			(1)  v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217
(2)  v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218
(3)  v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = A190509
(4)  v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415 (intersections), A356104 (reverse composites), A356218, A190509, A356220.

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 *)
Table[v[[u[[n]]]], {n, 1, z/4}]   (* A356217 *)
Table[v1[[u[[n]]]], {n, 1, z/4}]  (* A356218 *)
Table[v[[u1[[n]]]], {n, 1, z/4}]  (* A190509 *)
Table[v1[[u1[[n]]]], {n, 1, z/4}] (* A356220 *)

from math import isqrt
def A356217(n): return isqrt(5*(n+isqrt(5*n**2)>>1)**2) # Chai Wah Wu, Oct 14 2022

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

Clark Kimberling, Oct 02 2022

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

			(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

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

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 *)

A356220 a(n) = A108598(A001950(n)).

Original entry on oeis.org

3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, 61, 65, 70, 74, 79, 85, 88, 94, 97, 103, 108, 112, 117, 123, 126, 132, 135, 141, 146, 150, 155, 161, 164, 170, 173, 179, 184, 188, 193, 197, 202, 208, 211, 217, 222, 226, 231, 235, 240, 246, 249, 255, 258, 264

Offset: 1

Clark Kimberling, Nov 13 2022

This is the fourth of four sequences that partition the positive integers. See A356217.

			(1)  v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217
(2)  v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218
(3)  v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = A190509
(4)  v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220

Cf. A000201, A001950, A022839, A108598, A351415 (intersections), A356104 (reverse composites), A356217, A356218, A356219.

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[v[[u[[n]]]], {n, 1, z/4}]   (* A356217 *)
Table[v1[[u[[n]]]], {n, 1, z/4}]  (* A356218 *)
Table[v[[u1[[n]]]], {n, 1, z/4}]  (* A190509 *)
Table[v1[[u1[[n]]]], {n, 1, z/4}] (* A356220 *)

A356101 Intersection of A000201 and A022839.

Original entry on oeis.org

1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, 43, 45, 48, 50, 56, 59, 61, 63, 66, 72, 74, 77, 79, 85, 88, 90, 92, 95, 97, 101, 103, 106, 108, 110, 113, 119, 121, 124, 126, 132, 135, 137, 139, 142, 144, 148, 150, 153, 155, 161, 166, 168, 171, 173, 177, 179

Offset: 1

Clark Kimberling, Sep 04 2022

This is the second of four sequences that partition the positive integers. See A351415.

			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. For A351415, 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.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356102, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]   (* A351415 *)
Intersection[u, v1]  (* A356101 *)
Intersection[u1, v]  (* A356102 *)
Intersection[u1, v1] (* A356103 *)

A356102 Intersection of A001950 and A022839.

Original entry on oeis.org

2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, 89, 91, 96, 102, 107, 109, 120, 125, 136, 138, 143, 149, 154, 167, 172, 178, 183, 185, 196, 201, 212, 214, 219, 225, 230, 243, 248, 259, 261, 272, 277, 290, 295, 301, 306, 308, 319, 324, 326, 328, 330, 333, 335

Offset: 1

Clark Kimberling, Sep 04 2022

This is the third of four sequences that partition the positive integers. See A351415.

			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. For A351415, 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 that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356101, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]   (* A351415 *)
Intersection[u, v1]  (* A356101 *)
Intersection[u1, v]  (* A356102 *)
Intersection[u1, v1] (* A356103 *)

A356103 Intersection of A001950 and A108598.

Original entry on oeis.org

5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, 57, 65, 68, 70, 75, 81, 83, 86, 94, 99, 104, 112, 115, 117, 123, 128, 130, 133, 141, 146, 151, 157, 159, 162, 164, 170, 175, 180, 188, 191, 193, 198, 204, 206, 209, 217, 222, 227, 233, 235, 238, 240, 246, 251

Offset: 1

Clark Kimberling, Sep 04 2022

This is the fourth of four sequences that partition the positive integers. See A351415.

			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. For A351415, 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 that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356101, A356102, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]   (* A351415 *)
Intersection[u, v1]  (* A356101 *)
Intersection[u1, v]  (* A356102 *)
Intersection[u1, v1] (* A356103 *)

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

Clark Kimberling, Sep 08 2022

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

			(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

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

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 *)

A356218 a(n) = A108598(A000201(n)).

Original entry on oeis.org

1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, 37, 39, 43, 45, 48, 52, 54, 57, 59, 63, 66, 68, 72, 75, 77, 81, 83, 86, 90, 92, 95, 99, 101, 104, 106, 110, 113, 115, 119, 121, 124, 128, 130, 133, 137, 139, 142, 144, 148, 151, 153, 157, 159, 162, 166, 168, 171

Offset: 1

Clark Kimberling, Oct 02 2022

This is the second of four sequences that partition the positive integers. See A356217.

			(1)  v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217
(2)  v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218
(3)  v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = A190509
(4)  v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415 (intersections), A356104 (reverse composites), A356217, A190509, A356220.

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[v[[u[[n]]]], {n, 1, z/4}]   (* A356217 *)
Table[v1[[u[[n]]]], {n, 1, z/4}]  (* A356218 *)
Table[v[[u1[[n]]]], {n, 1, z/4}]  (* A190509 *)
Table[v1[[u1[[n]]]], {n, 1, z/4}] (* A356220 *)

cp's OEIS Frontend

A190509 a(n) = n + [nr/s] + [nt/s] + [nu/s] where r=golden ratio, s=r^2, t=r^3, u=r^4, and [] represents the floor function.

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A351415 Intersection of Beatty sequences for (1+sqrt(5))/2 and sqrt(5).

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A356217 a(n) = A022839(A000201(n)).

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

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A356220 a(n) = A108598(A001950(n)).

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A356101 Intersection of A000201 and A022839.

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A356102 Intersection of A001950 and A022839.

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A356103 Intersection of A001950 and A108598.

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