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

%I A259725 #12 Nov 15 2024 03:06:39
%S A259725 1,4,10,13,16,17,18,19,20,21,22,23,24,25,27,28,30,31,32,35,37,38,44,
%T A259725 47,50,51,53,54,57,60,61,63,64,66,69,73,76,78,79,80,81,83,85,86,88,90,
%U A259725 97,98,100,103,104,106,107,110,113,114,117,120,126,129,132,133
%N A259725 Numbers k such that [r[s*k]] = [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor.
%C A259725 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 A259725 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 A259725 G. C. Greubel, <a href="/A259725/b259725.txt">Table of n, a(n) for n = 1..1000</a>
%t A259725 z = 1000; r = Sqrt[2]; s = Sqrt[3];
%t A259725 u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
%t A259725 v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
%t A259725 Select[Range[400], u[[#]] < v[[#]] &]   (* A259724 *)
%t A259725 Select[Range[200], u[[#]] == v[[#]] &]  (* A259725 *)
%t A259725 Select[Range[200], u[[#]] > v[[#]] &]   (* A259726 *)
%Y A259725 Cf. A259724, A259746.
%K A259725 nonn,easy
%O A259725 1,2
%A A259725 _Clark Kimberling_, Jul 15 2015