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

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

Author

Clark Kimberling, Jul 15 2015

Keywords

• nonn
• easy

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

• G. C. Greubel, Table of n, a(n) for n = 1..1000

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 *)