A259724 Numbers k such that [r[s*k]] < [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor.

5, 8, 15, 29, 34, 39, 42, 45, 46, 49, 56, 58, 68, 71, 75, 87, 92, 95, 99, 102, 105, 109, 112, 116, 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

Offset

1,1

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. Conjecture: the limits g = lim G(n)/n, e = lim E(n)/n, el = lim L(n) exist; if so, then 1/g + 1/e + 1/el = 1.) In particular, A259724, A259725, A259726 partition the positive integers.

Links

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

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 *)

Crossrefs

Cf. A259725, A259746.

Sequence in context: A314559 A314560 A327605 * A259585 A220034 A063731

Adjacent sequences: A259721 A259722 A259723 * A259725 A259726 A259727

Keyword

nonn,easy

Author

Clark Kimberling, Jul 15 2015

Status

approved