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A024845 a(n) = least m such that if r and s in {1/2, 1/4, 1/6, ..., 1/(2*n)} satisfy r < s, then r < k/m < (k+3)/m < s for some integer k.

%I A024845 #28 Sep 01 2022 17:29:40
%S A024845 15,41,79,129,191,265,351,449,577,703,861,1013,1201,1379,1597,1801,
%T A024845 2049,2279,2557,2813,3121,3403,3741,4049,4417,4801,5151,5565,5995,
%U A024845 6385,6845,7321,7751,8257,8779,9249,9801,10369,10879,11477,12091,12641,13285,13945
%N A024845 a(n) = least m such that if r and s in {1/2, 1/4, 1/6, ..., 1/(2*n)} satisfy r < s, then r < k/m < (k+3)/m < s for some integer k.
%C A024845 For a guide to related sequences, see A001000. - _Clark Kimberling_, Aug 12 2012
%C A024845 From _Jianing Song_, Aug 31 2022: (Start)
%C A024845 Smallest m such that ceiling(m/(2*j)) - floor(m/(2*j+2)) >= 5 for 1 <= j <= n-1.
%C A024845 Obviously we have a(n) > 3/(1/(2*n-2) - 1/(2*n)) = 6*n*(n-1). On the other hand, a(n) <= 4/(1/(2*n-2) - 1/(2*n)) + 1 = 8*n*(n-1) + 1: if m >= 8*n*(n-1) + 1, then m/(2*j) - m/(2*j+2) > 4 => ceiling(m/(2*j)) - floor(m/(2*j+2)) = ceiling(m/(2*j)-floor(m/(2*j+2))) >= ceiling(m/(2*j) - m/(2*j+2)) >= 5 for 1 <= j <= n-1.
%C A024845 a(n) is odd for all n: for even m, we have ceiling(m/(2*j)) = ceiling((m-1)/(2*j)) (otherwise (m-1)/(2*j) would be an integer, which is impossible), so ceiling(m/(2*j)) - floor(m/(2*j+2)) >= 5 implies ceiling((m-1)/(2*j)) - floor((m-1)/(2*j+2)) >= 5. (End)
%H A024845 Clark Kimberling, <a href="/A024845/b024845.txt">Table of n, a(n) for n = 2..100</a>
%t A024845 leastSeparatorS[seq_, s_] := Module[{n = 1},
%t A024845 Table[While[Or @@ (Ceiling[n #1[[1]]] <
%t A024845 s + 1 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
%t A024845 Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
%t A024845 t = Map[leastSeparatorS[1/(2*Range[50]), #] &, Range[5]];
%t A024845 t[[4]] (* A024845 *)
%t A024845 (* _Peter J. C. Moses_, Aug 06 2012 *)
%o A024845 (PARI) a(n) = for(m=6*n^2-6*n+1, 8*n^2-8*n+1, forstep(j=n-1, 1, -1, if(-((-m)\(2*j)) - m\(2*j+2) < 5, break(), if(j==1, return(m))))) \\ _Jianing Song_, Aug 31 2022
%Y A024845 Cf. A001000, A024844.
%K A024845 nonn
%O A024845 2,1
%A A024845 _Clark Kimberling_