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%I A269573 #13 Feb 23 2018 22:02:58
%S A269573 2,4,23,4500,23314202,703143261541584,580028504455491926110281336263,
%T A269573 471554575224119231041268294704259548817134505334232514876247
%N A269573 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1,1,1,1,1,...)
%C A269573 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.
%H A269573 Clark Kimberling, <a href="/A269573/b269573.txt">Table of n, a(n) for n = 1..11</a>
%H A269573 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A269573 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A269573 (1/2)^(1/3) = 1/2 + 1/4 + 1/23 + ...
%t A269573 r[k_] := 1; f[x_, 0] = x; z = 10;
%t A269573 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A269573 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A269573 x = 2^(-1/3); Table[n[x, k], {k, 1, z}]  (* A269573 *)
%Y A269573 Cf. A269993 (guide to related sequences).
%K A269573 nonn,frac,easy
%O A269573 1,1
%A A269573 _Clark Kimberling_, Mar 15 2016

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