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

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

Author

Clark Kimberling, Jul 15 2015

Keywords

• nonn
• easy

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

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

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 *)