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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
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
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 12 2012
STATUS
approved