A259725 Numbers k such that [r[s*k]] = [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor.

1, 4, 10, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 35, 37, 38, 44, 47, 50, 51, 53, 54, 57, 60, 61, 63, 64, 66, 69, 73, 76, 78, 79, 80, 81, 83, 85, 86, 88, 90, 97, 98, 100, 103, 104, 106, 107, 110, 113, 114, 117, 120, 126, 129, 132, 133

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, A259746.

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 *)