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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.
4
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, 32, 32, 32, 32, 32, 32, 35, 35, 35, 39, 39, 39, 39, 55, 61, 61, 66, 68, 69, 69, 69, 70, 70, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71
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 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.
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[E*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