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A259725 Numbers k such that [r[s*k]] = [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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. A259724, A259746.

Sequence in context: A074939 A038464 A125966 * A260936 A043001 A191195

Adjacent sequences:  A259722 A259723 A259724 * A259726 A259727 A259728

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 15 2015

STATUS

approved

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Last modified September 28 17:19 EDT 2020. Contains 337393 sequences. (Running on oeis4.)