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%I #4 Jul 20 2013 14:07:11
%S 1,4,7,10,22,3,29,20,17,14,25,36,11,30,19,46,27,35,51,91,8,141,85,61,
%T 45,82,37,95,29,50,71,113,21,97,76,55,123,34,81,47,107,60,73,86,112,
%U 138,190,307,13,395,239,174,135,109,96,83,153,70,127,57,158,101
%N Numerator of least splitting rational of s(n) and s(n+1), where s(n) = 1 + 1/2^2 + ... + 1/n^2.
%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="/A227686/b227686.txt">Table of n, a(n) for n = 1..1000</a>
%e The denominators (A227685) and numerators (A227686) can be read from this chain: s(1) <= 1 < s(2) < 4/3 < s(3) < 7/5 < s(4) < 10/7 < s(5) < 22/15 < ...
%t Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = Sum[k^(-2), {k, 1, n}]
%t t = Table[r[s[n], s[n + 1]], {n, 1, 150}] (*fractions)
%t fd = Denominator[t] (*A227685*)
%t fn = Numerator[t] (*A227686*)
%Y Cf. A227631, A227685.
%K nonn,frac,easy
%O 1,2
%A _Clark Kimberling_, Jul 19 2013
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