A259587 -id:A259587 - cp's OEIS frontend

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

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

Clark Kimberling, Jul 15 2015

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.

Clark Kimberling, Table of n, a(n) for n = 1..10000

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

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[sk]] - [s[rk]] = -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

Clark Kimberling, Jul 15 2015

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, so that a(1) = 5.

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Clark Kimberling)

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

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[sk]] - [s[rk]] = 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

Clark Kimberling, Jul 15 2015

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.

G. C. Greubel, Table of n, a(n) for n = 1..10000

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

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 *)

cp's OEIS Frontend

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

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

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

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cp's OEIS Frontend

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

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

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Keywords

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Examples

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Mathematica

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

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Author

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Mathematica

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

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

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