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.

Showing 1-6 of 6 results.

A259725 Numbers k such that [r[s*k]] = [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

1, 4, 10, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 35, 37, 38, 44, 47, 50, 51, 53, 54, 57, 60, 61, 63, 64, 66, 69, 73, 76, 78, 79, 80, 81, 83, 85, 86, 88, 90, 97, 98, 100, 103, 104, 106, 107, 110, 113, 114, 117, 120, 126, 129, 132, 133
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

Suppose that r and s are distinct real numbers, and let f(r,s,k) = [s[r*k]] - [r[s*k]]. Let (G(n)) be the sequence of those k for which f(r,s,k) > 0, let (E(n)) be those for which f(r,s,k) = 0, and (L(n)), those for which f(r,s,k) < 0. Clearly (G(n), E(n), L(n)) partition the positive integers. In particular, A259724, A259725, A259726 partition the positive integers.
Conjecture: the limits g = lim G(n)/n, e = lim E(n)/n, el = lim L(n)/n exist; if so, then 1/g + 1/e + 1/el = 1.

Links

Crossrefs

Cf. A259724, A259746.

Programs

  • Mathematica
    z = 1000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Select[Range[400], u[[#]] < v[[#]] &]   (* A259724 *)
Select[Range[200], u[[#]] == v[[#]] &]  (* A259725 *)
Select[Range[200], u[[#]] > v[[#]] &]   (* A259726 *)

A259584 Numbers k such that [r[s*k]] - [s[r*k]] = -2, where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

116, 314, 512, 657, 1340, 1422, 1620, 1818, 1900, 2161, 2243, 2441, 2639, 2982, 3124, 3322, 3747, 3800, 3945, 4027, 4143, 4225, 4766, 5251, 5449, 5531, 5729, 5927, 6125, 6270, 6352, 6953, 7091, 7233, 7431, 7711, 7774, 7856, 8054, 8252, 8457, 8595, 9278, 9360
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

It is easy to prove that [r[s*k]] - [s[r*k]] ranges from -2 to 2. For k = 1 to 10, the values of [r[s*k]] - [s[r*k]] are 0, 1, 1, 0, -1, 1, 1, -1, 1, 0.
The first -2 occurs when k = 116.

Links

Crossrefs

Cf. A259585, A259586, A259587, A259724, A259725, A259746.

Programs

  • Mathematica
    z = 12000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Flatten[Position[u - v, -2]] (* A259584 *)
Take[Flatten[Position[u - v, -1]], 100] (* A259585 *)
Take[Flatten[Position[u - v, 0]], 100]  (* A259725 *)
Take[Flatten[Position[u - v, 1]], 100]  (* A259587 *)
Take[Flatten[Position[u - v, 2]], 100]  (* A259586 *)
Select[Range[10000],Floor[Sqrt[2]Floor[Sqrt[3]#]]-Floor[Sqrt[3]Floor[ Sqrt[ 2]#]]==-2&] (* Harvey P. Dale, Dec 01 2016 *)

A259585 Numbers k such that [r[s*k]] - [s[r*k]] = -1, where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

5, 8, 15, 29, 34, 39, 42, 45, 46, 49, 56, 58, 68, 71, 75, 87, 92, 95, 99, 102, 105, 109, 112, 121, 124, 127, 128, 131, 145, 150, 157, 162, 169, 174, 177, 184, 187, 191, 198, 203, 206, 213, 232, 237, 240, 243, 244, 247, 254, 256, 266, 269, 273, 285, 290, 295
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

It is easy to prove that [r[s*k]] - [s[r*k]] ranges from -2 to 2.

Examples

			For k = 1 to 10, the values of [r[s*k]] - [s[r*k]] are 0, 1, 1, 0, -1, 1, 1, -1, 1, 0, so that a(1) = 5.

Links

Crossrefs

Cf. A259584, A259586, A259587, A259724, A259725, A259746.

Programs

  • Mathematica
    z = 12000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Flatten[Position[u - v, -2]] (* A259584 *)
Take[Flatten[Position[u - v, -1]], 100] (* A259585 *)
Take[Flatten[Position[u - v, 0]], 100]  (* A259725 *)
Take[Flatten[Position[u - v, 1]], 100]  (* A259587 *)
Take[Flatten[Position[u - v, 2]], 100]  (* A259586 *)

A259586 Numbers k such that [r[s*k]] - [s[r*k]] = 2, where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

41, 67, 70, 123, 130, 205, 212, 328, 350, 403, 410, 444, 526, 548, 555, 608, 671, 700, 724, 750, 753, 806, 869, 888, 898, 951, 1026, 1033, 1067, 1086, 1096, 1149, 1224, 1231, 1265, 1291, 1294, 1347, 1376, 1429, 1489, 1504, 1545, 1571, 1574, 1627, 1709, 1716
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

It is easy to prove that [r[s*k]] - [s[r*k]] ranges from -2 to 2. For k = 1 to 10, the values of [r[s*k]] - [s[r*k]] are 0, 1, 1, 0, -1, 1, 1, -1, 1, 0; the first appearance of 2 is when k = 41.

Links

Crossrefs

Cf. A259584, A259585, A259587, A259724, A259725, A259746.

Programs

  • Mathematica
    z = 12000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Flatten[Position[u - v, -2]] (* A259584 *)
Take[Flatten[Position[u - v, -1]], 100] (* A259585 *)
Take[Flatten[Position[u - v, 0]], 100]  (* A259725 *)
Take[Flatten[Position[u - v, 1]], 100]  (* A259587 *)
Take[Flatten[Position[u - v, 2]], 100]  (* A259586 *)
Select[Range[2000],Floor[Sqrt[2]Floor[Sqrt[3]#]]-Floor[Sqrt[3]Floor[Sqrt[2]#]]==2&] (* Harvey P. Dale, Aug 10 2024 *)

A259587 Numbers k such that [r[s*k]] - [s[r*k]] = 2, where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

2, 3, 6, 7, 9, 11, 12, 14, 26, 33, 36, 40, 43, 48, 52, 55, 59, 62, 65, 72, 74, 77, 82, 84, 89, 91, 93, 94, 96, 101, 108, 111, 115, 118, 119, 122, 125, 134, 137, 140, 141, 144, 147, 148, 149, 151, 152, 154, 159, 164, 171, 175, 178, 181, 188, 190, 193, 194
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

It is easy to prove that [r[s*k]] - [s[r*k]] ranges from -2 to 2. For k = 1 to 10, the values of [r[s*k]] - [s[r*k]] are 0, 1, 1, 0, -1, 1, 1, -1, 1, 0; the first appearance of 2 is when k = 41.

Links

Crossrefs

Cf. A259584, A259585, A259586, A259724, A259725, A259746.

Programs

  • Mathematica
    z = 12000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Flatten[Position[u - v, -2]] (* A259584 *)
Take[Flatten[Position[u - v, -1]], 100] (* A259585 *)
Take[Flatten[Position[u - v, 0]], 100]  (* A259725 *)
Take[Flatten[Position[u - v, 1]], 100]  (* A259587 *)
Take[Flatten[Position[u - v, 2]], 100]  (* A259586 *)

A259726 Numbers k such that [r[s*k]] > [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

2, 3, 6, 7, 9, 11, 12, 14, 26, 33, 36, 40, 41, 43, 48, 52, 55, 59, 62, 65, 67, 70, 72, 74, 77, 82, 84, 89, 91, 93, 94, 96, 101, 108, 111, 115, 118, 119, 122, 123, 125, 130, 134, 137, 140, 141, 144, 147, 148, 149, 151, 152, 154, 159, 164, 171, 175, 178, 181
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

Suppose that r and s are distinct real numbers, and let f(r,s,k) = [s[r*k]] - [r[s*k]]. Let (G(n)) be the sequence of those k for which f(r,s,k) > 0, let (E(n)) be those for which f(r,s,k) = 0, and (L(n)), those for which f(r,s,k) < 0. Clearly (G(n), E(n), L(n)) partition the positive integers. In particular, A259724, A259725, A259726 partition the positive integers.
Conjecture: the limits g = lim G(n)/n, e = lim E(n)/n, el = lim L(n)/n exist; if so, then 1/g + 1/e + 1/el = 1.

Links

Crossrefs

Cf. A259724, A259745.

Programs

  • Mathematica
    z = 1000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Select[Range[400], u[[#]] < v[[#]] &]  (* A259724 *)
Select[Range[200], u[[#]] == v[[#]] &] (* A259725 *)
Select[Range[200], u[[#]] > v[[#]] &]  (* A259726 *)
Showing 1-6 of 6 results.

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