A356088 -id:A356088 - cp's OEIS frontend

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

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)).

A356091 a(n) = A001952(A054406(n)).

Original entry on oeis.org

6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, 95, 102, 112, 119, 126, 136, 143, 150, 160, 167, 177, 184, 191, 201, 208, 215, 225, 232, 238, 249, 256, 266, 273, 279, 290, 297, 303, 314, 320, 331, 338, 344, 355, 361, 368, 378, 385, 392, 402, 409, 419, 426, 433

Offset: 1

Clark Kimberling, Aug 05 2022

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

			(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

Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356088, A356089, A356090.

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 *)

A356085 Intersection of A001951 and A054406.

Original entry on oeis.org

2, 4, 7, 9, 11, 14, 16, 18, 21, 26, 28, 33, 35, 42, 49, 52, 56, 59, 63, 66, 70, 73, 80, 82, 87, 89, 94, 97, 101, 104, 106, 108, 111, 113, 115, 118, 120, 123, 125, 127, 130, 132, 134, 137, 141, 144, 149, 151, 156, 158, 165, 172, 175, 179, 182, 186, 189, 196

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 A346308.

			(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

Cf. A001951, A001952, A022838, A054406, A346308, 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 *)

A356086 Intersection of A001952 and A022838.

Original entry on oeis.org

3, 6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, 81, 88, 95, 102, 105, 109, 112, 116, 119, 122, 126, 129, 133, 136, 143, 150, 157, 174, 180, 187, 204, 211, 218, 221, 225, 228, 232, 235, 242, 245, 249, 252, 256, 259, 266, 273, 284, 285, 287, 289, 290, 292, 294

Offset: 1

Clark Kimberling, Aug 04 2022

This is the third of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A346308.

			(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, ...) = A356085
(3)  u' ^ v  = ( 3,  6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, ...) = A356086
(4)  u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087

Cf. A001951, A001952, A022838, A054406, A346308, A356085, A356087, A356088 (results of composition 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 *)
Intersection[u, v]    (* A346308 *)
Intersection[u, v1]   (* A356085 *)
Intersection[u1, v]   (* A356086 *)
Intersection[u1, v1]  (* A356087 *)

from math import isqrt
from itertools import count, islice
def A356086_gen(): # generator of terms
    return filter(lambda n:n == isqrt(3*(isqrt(n**2//3)+1)**2),((k:=n<<1)+isqrt(k*n) for n in count(1)))
A356086_list = list(islice(A356086_gen(),30)) # Chai Wah Wu, Aug 06 2022

A356087 Intersection of A001952 and A054406.

Original entry on oeis.org

23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, 92, 99, 139, 146, 153, 160, 163, 167, 170, 177, 184, 191, 194, 198, 201, 208, 215, 238, 262, 269, 276, 279, 283, 286, 288, 291, 293, 295, 298, 300, 302, 305, 307, 309, 312, 314, 317, 319, 321, 324, 326, 328

Offset: 1

Clark Kimberling, Aug 04 2022

This is the fourth of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A346308.

			(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, ...) = A356085.
(3)  u' ^ v  = ( 3,  6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, ...) = A356086.
(4)  u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087.

Cf. A001951, A001952, A022838, A054406, A346308, A356085, A356086, A356088 (results of composition instead of intersections).

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 *)
Intersection[u, v]    (* A346308 *)
Intersection[u, v1]   (* A356085 *)
Intersection[u1, v]   (* A356086 *)
Intersection[u1, v1]  (* A356087 *)

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

Clark Kimberling, Aug 04 2022

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

			(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.

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

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

Clark Kimberling, Aug 04 2022

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

			(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

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

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

Clark Kimberling, Aug 24 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 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.

			(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

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

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 *)

from math import isqrt
def A356180(n): return isqrt(3*isqrt(n**2<<1)**2) # Chai Wah Wu, Sep 06 2022

A356181 a(n) = A054406(A001951(n)).

Original entry on oeis.org

2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, 49, 52, 56, 59, 61, 66, 68, 73, 75, 78, 82, 85, 89, 92, 97, 99, 101, 106, 108, 113, 115, 118, 123, 125, 130, 132, 134, 139, 141, 146, 149, 153, 156, 158, 163, 165, 170, 172, 175, 179, 182, 186, 189, 194

Offset: 1

Clark Kimberling, Aug 24 2022

This is the second of four sequences that partition the positive integers. See A356180.

			(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

Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356180, A356182, A356183.

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 *)

A356182 a(n) = A022838(A001952(n)).

Original entry on oeis.org

5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, 81, 88, 93, 100, 105, 110, 117, 122, 129, 135, 140, 147, 152, 159, 164, 171, 176, 181, 188, 193, 200, 206, 211, 218, 223, 230, 235, 240, 247, 252, 259, 265, 271, 277, 282, 289, 294, 301, 306, 311, 318, 323

Offset: 1

Clark Kimberling, Aug 24 2022

This is the third of four sequences that partition the positive integers. See A356180.

			(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

Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356180, A356181, A356183.

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 *)

from math import isqrt
def A356182(n): return isqrt(3*((k:=n<<1)+isqrt(k*n))**2) # Chai Wah Wu, Sep 05 2022

cp's OEIS Frontend

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

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Formula

A356091 a(n) = A001952(A054406(n)).

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A356085 Intersection of A001951 and A054406.

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A356086 Intersection of A001952 and A022838.

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A356087 Intersection of A001952 and A054406.

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

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

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

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A356181 a(n) = A054406(A001951(n)).

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