A024840 a(n) = least m such that if r and s in {1/1, 1/2, 1/3, ..., 1/n} satisfy r < s, then r < k/m < (k+2)/m < s for some integer k.

7, 17, 31, 49, 71, 97, 127, 169, 209, 262, 311, 375, 433, 508, 575, 661, 737, 834, 919, 1027, 1141, 1241, 1366, 1497, 1611, 1753, 1901, 2029, 2188, 2353, 2495, 2671, 2853, 3009, 3202, 3401, 3571, 3781, 3997, 4219, 4409, 4642, 4881, 5126, 5335, 5591, 5853, 6121, 6349, 6628

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Comments

For a guide to related sequences, see A001000. - Clark Kimberling, Aug 08 2012

Examples

			Using the terminology introduced at A001000, the 3rd separator of the set {1/3, 1/2, 1} is a(3) = 17, since 1/3 < 6/17 < 8/17 < 1/2 < 8/17 < 10/17 < 1 and 17 is the least m for which 1/3, 1/2, 1 are thus separated using numbers k/m. - _Clark Kimberling_, Aug 08 2012
		

Links

• Clark Kimberling, Table of n, a(n) for n = 2..100

Crossrefs

Cf. A001000.

Programs

Mathematica

leastSeparatorS[seq_, s_] := Module[{n = 1},
Table[While[Or @@ (Ceiling[n #1[[1]]] <
s + 1 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
t = Map[leastSeparatorS[1/Range[50], #] &, Range[5]];
TableForm[t]
t[[3]] (* Peter J. C. Moses, Aug 08 2012 *)