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A259724
Numbers k such that [r[s*k]] < [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor.
7
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
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
Sequence in context: A314559 A314560 A327605 * A259585 A220034 A063731
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 15 2015
STATUS
approved