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%I #9 Aug 12 2013 19:42:33
%S 1,2,3,4,5,9,1,13,8,6,5,4,7,3,11,8,5,7,11,2,19,11,9,7,5,8,11,20,3,13,
%T 7,11,15,4,13,9,5,16,11,6,13,7,15,8,9,10,11,13,14,17,20,24,31,43,69,1,
%U 84,49,35,27,23,19,17,15,14,12,11,21,10,9,17,8,15
%N Least splitter of n-th and (n+1)st partial sums of 1/1 + 1/3 + ... + 1/(2n-1).
%C Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. It appears that c/d is an integer (i.e., d = 1) for rationals in positions given by A082315; e.g. 1, 7, 56, ...
%H Clark Kimberling, <a href="/A227778/b227778.txt">Table of n, a(n) for n = 1..1000</a>
%e The first 15 splitting rationals are 1/1, 3/2, 5/3, 7/4, 9/5, 17/9, 2/1, 27/13, 17/8, 13/6, 11/5, 9/4, 16/7, 7/3, 26/11.
%t z = 16; r[x_, y_] := Module[{a, b, x1 = Min[{x, y}], y1 = Max[{x, y}]}, If[x == y, x, b = NestWhile[#1 + 1 &, 1, ! (a = Ceiling[#1 x1 - 1]) < Ceiling[#1 y1] - 1 &]; (a + 1)/b]]; s[n_] := s[n] = Sum[1/(k - 1)!, {k, 1, n}]; N[Table[s[k], {k, 1, z}]]; t = Table[r[s[n], s[n + 1]], {n, 2, z}]; Denominator[t] (* _Peter J. C. Moses_, Jul 15 2013 *)
%Y Cf. A227631.
%K nonn,frac,easy
%O 1,2
%A _Clark Kimberling_, Jul 30 2013
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