A356056 -id:A356056 - 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]]

A356052 Intersection of A001951 and A137803.

Original entry on oeis.org

1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, 32, 36, 38, 42, 45, 49, 53, 55, 57, 59, 63, 65, 66, 70, 72, 74, 76, 80, 82, 84, 86, 89, 91, 93, 97, 101, 103, 107, 111, 114, 118, 120, 124, 128, 130, 132, 135, 137, 141, 145, 147, 149, 151, 155, 156, 158, 162, 164

Offset: 1

Clark Kimberling, Jul 26 2022

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.
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 A356052, 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 ^ 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, A356053, A356054, 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 *)

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 *)

A356138 a(n) = A137803(A001951(n)).

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 17, 21, 22, 26, 28, 30, 34, 36, 40, 42, 45, 47, 49, 53, 55, 59, 61, 63, 66, 68, 72, 74, 78, 80, 82, 86, 88, 91, 93, 95, 99, 101, 105, 107, 109, 112, 114, 118, 120, 124, 126, 128, 132, 133, 137, 139, 141, 145, 147, 151, 153, 156, 158, 160

Offset: 1

Clark Kimberling, Aug 06 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 A356056 to A356059.
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 A356138, 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)  v o u   = (1,  3,  7,  9, 13, 15, 17, 21, 22, 26, 28, 30, 34, ...) = A356138
(2)  v' o u  = (2,  4,  8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, ...) = A356139
(3)  v o u'  = (5, 11, 19, 24, 32, 38, 44, 51, 57, 65, 70, 76, 84, ...) = A356140
(4)  v' o u' = (6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, ...) = A356141

Cf. A001951, A001952, A137804.

Cf. A356056, A356057, A356058, A356059, A356139, A356140, A356141.

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[v[[u[[n]]]], {n, 1, z/8}]   (* A356138 *)
Table[v1[[u[[n]]]], {n, 1, z/8}]  (* A356139 *)
Table[v[[u1[[n]]]], {n, 1, z/8}]  (* A356140 *)
Table[v1[[u1[[n]]]], {n, 1, z/8}] (* A356141 *)

A356139 a(n) = A137804(A001951(n)).

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, 39, 43, 46, 50, 52, 54, 58, 60, 64, 67, 69, 73, 75, 79, 81, 85, 87, 90, 94, 96, 100, 102, 104, 108, 110, 115, 117, 119, 123, 125, 129, 131, 136, 138, 140, 144, 146, 150, 152, 154, 159, 161, 165, 167, 171, 173

Offset: 1

Clark Kimberling, Aug 06 2022

This is the second of four sequences that partition the positive integers. See A356138.

			(1)  v o u   = (1,  3,  7,  9, 13, 15, 17, 21, 22, 26, 28, 30, 34, ...) = A356138
(2)  v' o u  = (2,  4,  8, 10, 14, 16, 18, 23, 25, 29, 31, 33, 37, ...) = A356139
(3)  v o u'  = (5, 11, 19, 24, 32, 38, 44, 51, 57, 65, 70, 76, 84, ...) = A356140
(4)  v' o u' = (6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, ...) = A356141

Cf. A001951, A001952, A137804.

Cf. A356056, A356057, A356058, A356059, A356138, A356140, A356141.

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[v[[u[[n]]]], {n, 1, z/8}]   (*A356138 *)
Table[v1[[u[[n]]]], {n, 1, z/8}]  (* A356139*)
Table[v[[u1[[n]]]], {n, 1, z/8}]  (* A356140 *)
Table[v1[[u1[[n]]]], {n, 1, z/8}] (* A356141 *)

cp's OEIS Frontend

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

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A356052 Intersection of A001951 and A137803.

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

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

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

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A356053 Intersection of A001951 and A137804.

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A356054 Intersection of A001952 and A137803.

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A356055 Intersection of A001952 and A137804.

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A356138 a(n) = A137803(A001951(n)).

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