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

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

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 (terms 1..1000 from Clark Kimberling)

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 *)