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Least splitter of the harmonic numbers H(n) and H(n+1).
3

%I #7 Dec 04 2016 19:46:32

%S 1,2,1,4,3,2,3,4,6,1,10,6,4,7,3,5,9,2,7,5,3,7,4,5,6,7,10,14,27,1,18,

%T 12,9,7,6,5,9,4,11,7,13,3,11,8,5,7,9,13,25,2,15,9,7,12,5,8,11,17,3,13,

%U 10,7,15,4,13,9,5,11,17,6,7,15,8,9,11,12,14,17

%N Least splitter of the harmonic numbers H(n) and H(n+1).

%C See A227631 for the definition of least splitter.

%H Clark Kimberling, <a href="/A227629/b227629.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. A227630, A227631.

%K nonn,frac

%O 1,2

%A _Clark Kimberling_, Jul 18 2013