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A227687 Least splitter of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).

%I A227687 #7 Jun 24 2015 18:38:39
%S A227687 1,1,2,1,2,1,3,2,1,4,2,3,1,3,2,3,1,3,2,3,1,5,3,2,3,1,5,3,2,3,5,1,4,3,
%T A227687 2,3,5,1,4,3,2,3,4,1,6,4,3,2,3,5,1,6,4,3,2,3,4,7,1,5,3,5,2,3,4,7,1,5,
%U A227687 4,3,2,3,4,6,1,7,4,3,5,2,3,4,6,1,8,5
%N A227687 Least splitter of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).
%C A227687 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.
%C A227687 The positions of 1 in this sequences (indicating those least splitting rationals of s(n) and s(n+1) which are integers) are given by A186351.
%H A227687 Clark Kimberling, <a href="/A227687/b227687.txt">Table of n, a(n) for n = 1..1000</a>
%e A227687 The denominators (A227687) and numerators (A227688) can be read from these chains:
%e A227687 1 < 2 < 5/2 < 3 < 7/2 < 4 < 13/3 < 9/2 < 5 < 21/4 < 11/2 < 17/3 < 6 < . . .
%e A227687 s(1) <= 1 < s(2) < 2 < s(3) < 5/2 < s(4) < 3 < s(5) < 4 < s(6) < 13/3 <  . . .
%t A227687 r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d];
%t A227687 s[n_] := s[n] = Sum[k^(-1/2), {k, 1, n}]; t = Table[r[s[n], s[n + 1]], {n, 1, 15}] (*fractions*)
%t A227687 fd = Denominator[t] (*A227687*)
%t A227687 fn = Numerator[t]   (*A227688*)
%Y A227687 Cf. A227631, A227688.
%K A227687 nonn,frac,easy
%O A227687 1,3
%A A227687 _Clark Kimberling_, Jul 21 2013