

A214921


Least m > 0 such that for every r and s in the set S = {{h*sqrt(2))} : h = 1,..,n} of fractional parts, if r < s, then r < k/m < s for some integer k.


2



2, 3, 4, 5, 7, 7, 12, 12, 12, 12, 12, 15, 15, 17, 17, 17, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 33, 36, 36, 36, 41, 41, 41, 41, 41, 41, 41, 41, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70
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OFFSET

2,1


COMMENTS

a(n) is the least separator of S, as defined at A001000, which includes a guide to related sequences.  Clark Kimberling, Aug 12 2012


LINKS

Clark Kimberling, Table of n, a(n) for n = 2..300


EXAMPLE

Write the fractional parts of h*sqrt(2) for h=1,2,...,6, sorted, as f1, f2, f3, f4, f5, f6. Then f1 < 1/7 < f2 < 2/7 < f3 < 3/7 < f4 < 4/7 < f5 < 5/7 < f6, and 7 is the least m for which such a separation by fractions k/m occurs, so that a(6)=7.


MATHEMATICA

leastSeparatorShort[seq_, s_] := Module[{n = 1},
While[Or @@ (n #1[[1]] <= s + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[seq, 2, 1], n++]; n];
Table[leastSeparatorShort[Sort[N[FractionalPart[Sqrt[2] Range[n]], 50]], 1], {n, 2, 100}]
(* Peter J. C. Moses, Aug 01 2012 *)


CROSSREFS

Cf. A001000, A214964, A214965.
Sequence in context: A266620 A222415 A022473 * A265566 A265550 A306012
Adjacent sequences: A214918 A214919 A214920 * A214922 A214923 A214924


KEYWORD

nonn


AUTHOR

Clark Kimberling, Aug 12 2012


STATUS

approved



