cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

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

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

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

Examples

			(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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, Aug 06 2022

Keywords

Comments

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

Examples

			(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

Crossrefs

Cf. A001951, A001952, A137804.
Cf. A356056, A356057, A356058, A356059, A356138, A356140, A356141.

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, Aug 06 2022

Keywords

Comments

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

Examples

			(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

Crossrefs

Cf. A001951, A001952, A137804.
Cf. A356056, A356057, A356058, A356059, A356138, A356139, A356140.

Programs

  • Mathematica
    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 *)

A137805 Self-inverse integer permutation induced by Beatty sequences for Sqrt(2)+1/2 and (4*Sqrt(2)+9)/7.

Original entry on oeis.org

2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 18, 17, 20, 19, 23, 25, 21, 27, 22, 29, 24, 31, 26, 33, 28, 35, 30, 37, 32, 39, 34, 41, 36, 43, 38, 46, 40, 48, 50, 42, 52, 44, 54, 45, 56, 47, 58, 49, 60, 51, 62, 53, 64, 55, 67, 57, 69, 59, 71, 73, 61, 75, 63, 77, 65, 79
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 11 2008

Keywords

Links

Formula

a(A137803(n)) = A137804(n) and a(A137804(n)) = A137803(n).

A325733 First term of n-th difference sequence of (floor(k*r)), r = 1/2 + sqrt(2), k >= 0.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, 11, -77, 363, -1364, 4367, -12375, 31823, -75581, 167959, -352715, 705431, -1352078, 2496167, -4457699, 7728759, -13055444, 21572459, -35072309, 56663099, -92561039, 157073279, -286097759, 572195518, -1251677664
Offset: 1

Views

Author

Clark Kimberling, May 20 2019

Keywords

Links

Crossrefs

Cf. A325664. Inverse binomial transform of A137803.

Programs

  • Mathematica
    Table[First[Differences[Table[Floor[(1/2+Sqrt[2])*n], {n, 0, 50}], n]], {n, 1, 50}]

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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]]

A194148 Sum_{j=1..n} floor(j*(1/2 + sqrt(2))); n-th partial sum of Beatty sequence for 1/2 + sqrt(2).

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 143, 167, 193, 221, 251, 283, 317, 353, 391, 431, 473, 517, 562, 609, 658, 709, 762, 817, 874, 933, 994, 1057, 1122, 1188, 1256, 1326, 1398, 1472, 1548, 1626, 1706, 1788, 1872, 1958, 2046, 2135, 2226, 2319
Offset: 1

Views

Author

Clark Kimberling, Aug 17 2011

Keywords

Links

Crossrefs

Cf. A137803 (Beatty sequence for 1/2 + sqrt(2)).

Programs

  • Magma
    [(&+[Floor(k*(Sqrt(2) + 1/2)): k in [1..n]]): n in [1..60]]; // G. C. Greubel, Oct 05 2018
  • Mathematica
    c[n_] := Sum[Floor[j*(1/2+Sqrt[2])], {j, 1, n}];
c = Table[c[n], {n, 1, 90}]
Accumulate[Table[Floor[n(1/2+Sqrt[2])],{n,50}]] (* Harvey P. Dale, May 26 2023 *)
  • PARI
    for(n=1,60, print1(sum(j=1,n, floor(j*(sqrt(2) + 1/2))), ", ")) \\ G. C. Greubel, Oct 05 2018
Previous Showing 11-17 of 17 results.

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