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A024843 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+3)/m < s for some integer k.

%I A024843 #19 Dec 14 2021 10:30:20
%S A024843 9,23,43,69,101,139,183,233,289,361,431,518,601,703,799,916,1025,1157,
%T A024843 1279,1426,1561,1723,1871,2048,2209,2401,2601,2783,2998,3221,3423,
%U A024843 3661,3907,4129,4390,4659,4901,5185,5477,5739,6046,6361,6643,6973,7311,7613,7966,8327,8649
%N A024843 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+3)/m < s for some integer k.
%C A024843 For a guide to related sequences, see A001000. - _Clark Kimberling_, Aug 08 2012
%H A024843 Clark Kimberling, <a href="/A024843/b024843.txt">Table of n, a(n) for n = 2..100</a>
%e A024843 Using the terminology introduced at A001000, the 4th separator of the set {1/3, 1/2, 1} is a(3) = 23, since 1/3 < 8/23 < 11/23 < 1/2 < 12/23 < 15/23 < 1 and 23 is the least m for which 1/3, 1/2, 1 are thus separated using numbers k/m. - _Clark Kimberling_, Aug 08 2012
%t A024843 leastSeparatorS[seq_, s_] := Module[{n = 1},
%t A024843 Table[While[Or @@ (Ceiling[n #1[[1]]] <
%t A024843 s + 1 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
%t A024843 Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
%t A024843 t = Map[leastSeparatorS[1/Range[50], #] &, Range[5]];
%t A024843 TableForm[t]
%t A024843 t[[4]] (* _Peter J. C. Moses_, Aug 08 2012 *)
%Y A024843 Cf. A001000.
%K A024843 nonn
%O A024843 2,1
%A A024843 _Clark Kimberling_