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A024846 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+4)/m < s for some integer k.

%I A024846 #18 Dec 14 2021 10:30:33
%S A024846 11,29,55,89,131,181,239,305,379,461,551,661,769,898,1023,1171,1313,
%T A024846 1480,1639,1825,2001,2206,2399,2623,2833,3076,3303,3565,3809,4090,
%U A024846 4351,4651,4961,5249,5578,5917,6231,6589,6957,7297,7684,8081,8447,8863,9289,9681,10126
%N A024846 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+4)/m < s for some integer k.
%C A024846 For a guide to related sequences, see A001000. - _Clark Kimberling_, Aug 08 2012
%H A024846 Clark Kimberling, <a href="/A024846/b024846.txt">Table of n, a(n) for n = 2..100</a>
%e A024846 Using the terminology introduced at A001000, the 5th separator of the set {1/3, 1/2, 1} is a(3) = 29, since 1/3 < 10/29 < 14/29 < 1/2 < 15/29 < 19/29 < 1, and 29 is the least m for which 1/3, 1/2, 1 are thus separated using numbers k/m. - _Clark Kimberling_, Aug 08 2012
%t A024846 leastSeparatorS[seq_, s_] := Module[{n = 1},
%t A024846 Table[While[Or @@ (Ceiling[n #1[[1]]] <
%t A024846 s + 1 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
%t A024846 Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
%t A024846 t = Map[leastSeparatorS[1/Range[50], #] &, Range[5]];
%t A024846 TableForm[t]
%t A024846 t[[5]] (* _Peter J. C. Moses_, Aug 08 2012 *)
%Y A024846 Cf. A001000.
%K A024846 nonn
%O A024846 2,1
%A A024846 _Clark Kimberling_