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.

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

Author

Clark Kimberling, Aug 12 2012

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A001000, A214961, A214964.

Programs

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 *)