%I #6 Feb 23 2018 11:08:31
%S 2,2,4,21,168,10754,25461498,105205312405537,
%T 2273436544813042470905435068,
%U 580632014636885174037652548241171956049642213022500047,105076738483143967759563061000636154401568577693011463452250666394865203888381724797435152416096091560375615
%N Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/k!.
%C Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1)) + r(2)/(n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.
%C See A269993 for a guide to related sequences.
%H Clark Kimberling, <a href="/A270527/b270527.txt">Table of n, a(n) for n = 1..13</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e (1/2)^(1/3) = 1/(1*2) + 1/(2*2) + 1/(6*4) + 1/(24*21) + ...
%t r[k_] := 1/k!; f[x_, 0] = x; z = 10;
%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t x = (1/2)^(1/3); Table[n[x, k], {k, 1, z}]
%Y Cf. A269993, A000142.
%K nonn,frac,easy
%O 1,1
%A _Clark Kimberling_, Apr 02 2016
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