A356052 -id:A356052 - cp's OEIS frontend

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

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

Clark Kimberling, Jul 26 2022

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

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

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]]

A346308 Intersection of Beatty sequences for sqrt(2) and sqrt(3).

Original entry on oeis.org

1, 5, 8, 12, 15, 19, 22, 24, 25, 29, 31, 32, 36, 38, 39, 41, 43, 45, 46, 48, 50, 53, 55, 57, 60, 62, 65, 67, 69, 72, 74, 76, 77, 79, 83, 84, 86, 90, 91, 93, 96, 98, 100, 103, 107, 110, 114, 117, 121, 124, 128, 131, 135, 138, 140, 142, 145, 147, 148, 152, 154

Offset: 1

Clark Kimberling, Sep 11 2021

nonn

Let d(n) = a(n) - A022840(n). Conjecture: (d(n)) is unbounded below and above, and d(n) = 0 for infinitely many n.
From Clark Kimberling, Jul 26 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 A346308, 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. (See A356052.)
(End)

			Beatty sequence for sqrt(2): (1,2,4,5,7,8,9,11,12,14,...).
Beatty sequence for sqrt(3): (1,3,5,6,8,10,12,13,15,...).
a(n) = (1,5,8,12,...).
In the notation in Comments:
(1)  u ^ v = (1, 5, 8, 12, 15, 19, 22, 24, 25, 29, 31, 32, ...) =  A346308.
(2)  u ^ v' = (2, 4, 7, 9, 11, 14, 16, 18, 21, 26, 28, 33, 35, ...) =  A356085.
(3)  u' ^ v = (3, 6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, 81, ...) = A356086.
(4)  u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087.

Intersection of A001951 and A022838.
Cf. A346308, A347467, A347468, A347469.
Cf. A001952, A022838, A054406, A356085, A356086, A356087, A356088 (composites instead of intersections).

z = 200;
r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
r1 = Sqrt[3]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022838 *)
v1 = Take[Complement[Range[1000], v], z]  (* A054406 *)
t1 = Intersection[u, v]    (* A346308 *)
t2 = Intersection[u, v1]   (* A356085 *)
t3 = Intersection[u1, v]   (* A356086 *)
t4 = Intersection[u1, v1]  (* A356087 *)

from math import isqrt
from itertools import count, islice
def A346308_gen(): # generator of terms
    return filter(lambda n:n == isqrt(3*(isqrt(n**2//3)+1)**2),(isqrt(n*n<<1) for n in count(1)))
A346308_list = list(islice(A346308_gen(),30)) # Chai Wah Wu, Aug 06 2022

In general, if r and s are irrational numbers greater than 1, and a(n) is the n-th term of the intersection (assumed nonempty) of the Beatty sequences for r and s, then a(n) = floor(r*ceiling(a(n)/r)) = floor(s*ceiling(a(n)/s)).

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

Clark Kimberling, Jul 26 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) 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.

			(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

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356057, A356058, A356059.

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 *)

a(n) = A001951(A137803(n)).

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

Clark Kimberling, Jul 26 2022

This is the second of four sequences that partition the positive integers. See A356056.

			(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

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356056, A356058, A356059.

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 *)

a(n) = A001951(A137804(n)).

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

Clark Kimberling, Jul 26 2022

This is the third of four sequences that partition the positive integers. See A356056.

			(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

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of the results of composition), A356056, A356057, A356059.

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 *)

a(n) = A001952(A137803(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

Clark Kimberling, Jul 26 2022

This is the fourth of four sequences that partition the positive integers. See A356056.

			(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

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356056, A356057, A356058.

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 *)

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

Clark Kimberling, Jul 26 2022

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

			(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

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

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 *)

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

Clark Kimberling, Jul 26 2022

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

			(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

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

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 *)

A356055 Intersection of A001952 and A137804.

Original entry on oeis.org

6, 10, 20, 23, 27, 37, 54, 58, 64, 71, 75, 81, 85, 92, 102, 119, 129, 136, 146, 150, 157, 163, 167, 177, 180, 184, 194, 198, 201, 211, 215, 221, 228, 232, 238, 242, 249, 259, 276, 286, 293, 297, 303, 307, 314, 320, 324, 341, 351, 355, 358, 368, 372, 378, 385

Offset: 1

Clark Kimberling, Jul 26 2022

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

			(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

Cf. A001951, A001952, A136803, A137804, A356052, A356053, A356055, A356056 (composites instead of intersections), A356081.

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 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A346308 Intersection of Beatty sequences for sqrt(2) and sqrt(3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A356056 a(n) = A001951(A137803(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A356057 a(n) = A001951(A137804(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A356059 a(n) = A001952(A137804(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A356053 Intersection of A001951 and A137804.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356054 Intersection of A001952 and A137803.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356055 Intersection of A001952 and A137804.

Views