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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.

%I A214921 #13 Dec 25 2023 18:07:15
%S A214921 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,
%T A214921 29,29,29,33,36,36,36,41,41,41,41,41,41,41,41,70,70,70,70,70,70,70,70,
%U A214921 70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70,70
%N 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.
%C A214921 a(n) is the least separator of S, as defined at A001000, which includes a guide to related sequences. - _Clark Kimberling_, Aug 12 2012
%H A214921 Clark Kimberling, <a href="/A214921/b214921.txt">Table of n, a(n) for n = 2..300</a>
%e A214921 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.
%t A214921 leastSeparatorShort[seq_, s_] := Module[{n = 1},
%t A214921 While[Or @@ (n #1[[1]] <= s + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
%t A214921 Partition[seq, 2, 1], n++]; n];
%t A214921 Table[leastSeparatorShort[Sort[N[FractionalPart[Sqrt[2] Range[n]], 50]], 1], {n, 2, 100}]
%t A214921 (* _Peter J. C. Moses_, Aug 01 2012 *)
%Y A214921 Cf. A001000, A214964, A214965.
%K A214921 nonn
%O A214921 2,1
%A A214921 _Clark Kimberling_, Aug 12 2012