|
%I #5 Dec 04 2016 19:46:32
%S 1,3,2,9,7,5,8,11,17,3,31,19,13,23,10,17,31,7,25,18,11,26,15,19,23,27,
%T 39,55,107,4,73,49,37,29,25,21,38,17,47,30,56,13,48,35,22,31,40,58,
%U 112,9,68,41,32,55,23,37,51,79,14,61,47,33,71,19,62,43,24
%N Numerator of the least splitting rational of the harmonic numbers H(n) and H(n+1).
%C See A227631 for the definition of least splitting rational.
%H Clark Kimberling, <a href="/A227630/b227630.txt">Table of n, a(n) for n = 1..1000</a>
%e The first few splitting rationals are 1/1, 3/2, 2/1, 9/4, 7/3, 5/2, 8/3, 11/4, 17/6, 3/1, 31/10, 19/6; e.g. 9/4 splits H(4) and H(5), as indicated by H(4) = 1 + 1/2 + 1/3 + 1/4 = 2.083... < 2.25 < 2.283... = H(5) and the chain H(1) <= 1/1 < H(2) < 3/2 < H(3) < 2/1 < H(4) < 9/4 < ...
%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[h[n], h[n + 1]], {n, 1, 120}];
%t Denominator[t] (* A227629 *)
%t Numerator[t] (* A227630 *) (* _Peter J. C. Moses_, Jul 15 2013 *)
%Y Cf. A227629, A227631.
%K nonn,frac
%O 1,2
%A _Clark Kimberling_, Jul 18 2013
|
