cp's OEIS Frontend

A214965 Least m > 0 such that for every r and s in the set S = {{h*e} : h = 1,..,n} of fractional parts, if r < s, then r < k/m < s for some integer k; m is the least separator of S as defined at A001000.

%I A214965 #11 Mar 01 2014 09:43:38
%S A214965 2,3,4,6,6,7,11,11,11,18,18,18,18,25,25,25,25,25,25,25,25,25,32,32,32,
%T A214965 32,32,32,32,32,32,35,35,35,39,39,39,39,55,61,61,66,68,69,69,69,70,70,
%U A214965 71,71,71,71,71,71,71,71,71,71,71,71,71,71,71,71,71,71,71,71
%N A214965 Least m > 0 such that for every r and s in the set S = {{h*e} : h = 1,..,n} of fractional parts, if r < s, then r < k/m < s for some integer k; m is the least separator of S as defined at A001000.
%C A214965 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 A214965 Clark Kimberling, <a href="/A214965/b214965.txt">Table of n, a(n) for n = 2..300</a>
%e A214965 Write the sorted fractional parts {h*e}, for h=1..5, as f1,f2,f3,f4,f5.  Then f1 < 2/6 < f2 < 3/6 < f3 < 4/6 < f5 < 5/6 < f6, and no such separation occurs using fractions k/m having m < 6; so a(5) = 6.
%t A214965 leastSeparatorShort[seq_, s_] := Module[{n = 1},
%t A214965 While[Or @@ (n #1[[1]] <= s + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
%t A214965 Partition[seq, 2, 1], n++]; n];
%t A214965 Table[leastSeparatorShort[Sort[N[FractionalPart[E*Range[n]], 50]], 1], {n, 2, 100}]
%t A214965 (* _Peter J. C. Moses_, Aug 01 2012 *)
%Y A214965 Cf. A001000, A214961, A214964.
%K A214965 nonn
%O A214965 2,1
%A A214965 _Clark Kimberling_, Aug 12 2012