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

%I A259724 #13 Nov 15 2024 03:06:22
%S A259724 5,8,15,29,34,39,42,45,46,49,56,58,68,71,75,87,92,95,99,102,105,109,
%T A259724 112,116,121,124,127,128,131,145,150,157,162,169,174,177,184,187,191,
%U A259724 198,203,206,213,232,237,240,243,244,247,254,256,266,269,273,285,290
%N A259724 Numbers k such that [r[s*k]] < [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor.
%C A259724 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.
%C A259724 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.
%H A259724 G. C. Greubel, <a href="/A259724/b259724.txt">Table of n, a(n) for n = 1..1000</a>
%t A259724 z = 1000; r = Sqrt[2]; s = Sqrt[3];
%t A259724 u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
%t A259724 v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
%t A259724 Select[Range[400], u[[#]] < v[[#]] &]  (* A259724 *)
%t A259724 Select[Range[200], u[[#]] == v[[#]] &] (* A259725 *)
%t A259724 Select[Range[200], u[[#]] > v[[#]] &]  (* A259726 *)
%Y A259724 Cf. A259725, A259746.
%K A259724 nonn,easy
%O A259724 1,1
%A A259724 _Clark Kimberling_, Jul 15 2015