A137804 -id:A137804 - cp's OEIS frontend

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

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)).

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 *)

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 *)

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 *)

A356141 a(n) = A137804(A001952(n)).

Original entry on oeis.org

6, 12, 20, 27, 35, 41, 48, 56, 62, 71, 77, 83, 92, 98, 106, 113, 121, 127, 134, 142, 148, 157, 163, 169, 177, 184, 192, 198, 207, 213, 219, 228, 234, 242, 249, 255, 263, 270, 278, 284, 291, 299, 305, 314, 320, 328, 335, 341, 349, 355, 364, 370, 376, 385, 391

Offset: 1

Clark Kimberling, Aug 06 2022

This is the fourth 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, A356139, A356140.

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 *)

A137803 a(n) = floor(n*(sqrt(2) + 1/2)).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 112, 114, 116, 118, 120, 122, 124, 126

Offset: 1

Reinhard Zumkeller, Feb 11 2008

nonn

a(n) = A059533(n) for n <= 34;
Beatty sequence for sqrt(2) + 1/2; complement of A137804;
a(n) = A137805(A137804(n)) and A137805(a(n)) = A137804(n).

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from R. Zumkeller)
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A001951, A003151.

[Floor(n*(Sqrt(2)+1/2)): n in [1..50]]; // G. C. Greubel, Jan 27 2018

Floor[Range[80](Sqrt[2]+1/2)]  (* Harvey P. Dale, Mar 24 2011 *)

for(n=1,50, print1(floor(n*(sqrt(2)+1/2)), ", ")) \\ G. C. Greubel, Jan 27 2018

from math import isqrt
def A137803(n): return (n>>1)+(m:=isqrt(r:=n*n<<1))+(n&1)*int(r-m*(m+1)>=1) # Chai Wah Wu, Aug 03 2022

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)).

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 *)

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)).

cp's OEIS Frontend

A356057 a(n) = A001951(A137804(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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A356055 Intersection of A001952 and A137804.

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

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A356141 a(n) = A137804(A001952(n)).

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A137803 a(n) = floor(n*(sqrt(2) + 1/2)).

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

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A356052 Intersection of A001951 and A137803.