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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.

%I A024840 #19 Dec 14 2021 10:31:11
%S A024840 7,17,31,49,71,97,127,169,209,262,311,375,433,508,575,661,737,834,919,
%T A024840 1027,1141,1241,1366,1497,1611,1753,1901,2029,2188,2353,2495,2671,
%U A024840 2853,3009,3202,3401,3571,3781,3997,4219,4409,4642,4881,5126,5335,5591,5853,6121,6349,6628
%N 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.
%C A024840 For a guide to related sequences, see A001000. - _Clark Kimberling_, Aug 08 2012
%H A024840 Clark Kimberling, <a href="/A024840/b024840.txt">Table of n, a(n) for n = 2..100</a>
%e A024840 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
%t A024840 leastSeparatorS[seq_, s_] := Module[{n = 1},
%t A024840 Table[While[Or @@ (Ceiling[n #1[[1]]] <
%t A024840 s + 1 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
%t A024840 Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
%t A024840 t = Map[leastSeparatorS[1/Range[50], #] &, Range[5]];
%t A024840 TableForm[t]
%t A024840 t[[3]] (* _Peter J. C. Moses_, Aug 08 2012 *)
%Y A024840 Cf. A001000.
%K A024840 nonn
%O A024840 2,1
%A A024840 _Clark Kimberling_