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, 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

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

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

Cf. A001000, A214964, A214965.

Sequence in context: A266620 A222415 A022473 * A265566 A265550 A306012

Adjacent sequences: A214918 A214919 A214920 * A214922 A214923 A214924

Keyword

nonn

Author

Clark Kimberling, Aug 12 2012

Status

approved