A259584 Numbers k such that [r[s*k]] - [s[r*k]] = -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

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 -2 occurs when k = 116.

Links

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

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 *)