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%I #5 Jul 20 2013 14:06:29
%S 0,1,4,3,5,9,2,13,9,7,12,5,13,8,11,14,17,29,53,3,34,25,19,16,13,23,33,
%T 10,17,24,31,52,7,39,25,18,29,51,11,37,26,15,34,19,42,23,27,31,39,47,
%U 59,83,135,4,161,97,69,53,45,41,33,29,54,25,46,21,59,38
%N Numerator of least splitting rational of log(n) and log(n+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.
%H Clark Kimberling, <a href="/A227684/b227684.txt">Table of n, a(n) for n = 1..1000</a>
%e The splitting rationals of consecutive numbers log(1), log(2), ... are 0, 1, 4/3, 3/2, 5/3, 9/5, 2, 13/6, 9/4, 7/3, 12/5, 5/2, 13/5; the denominators form A227634, and the numerators, A227684. Chain:
%e log(1) <= 0 < log(2) < 1 < log(3) < 4/3 < log(4) < 3/2 < log(5) < 5/3 < ...
%t h[n_] := h[n] = HarmonicNumber[n]; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; t = Table[r[Log[n], Log[n + 1]], {n, 1, 120}] (*fractions*)
%t Denominator[t] (* A227634 *)
%t Numerator[t] (* A227684 *)
%Y Cf. A227631, A227634.
%K nonn,frac,easy
%O 1,3
%A _Clark Kimberling_, Jul 19 2013
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