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A270487 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/Prime(k).

%I A270487 #12 Feb 16 2025 08:33:31
%S A270487 1,2,2,6,29,860,626907,1582796431872,4577382865450526674426008,
%T A270487 77218331531088831524423800072197013265311322482652,
%U A270487 10410509369911993512345323774444196964795747018426948027297775848734862056109801420845614477793011811
%N A270487 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/Prime(k).
%C A270487 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 A270487 See A269993 for a guide to related sequences.
%H A270487 Clark Kimberling, <a href="/A270487/b270487.txt">Table of n, a(n) for n = 1..13</a>
%H A270487 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270487 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270487 (1/2)^(1/3) = 1/(2*1) + 1/(3*2) + 1/(5*2) + 1/(7*6) + ...
%t A270487 r[k_] := 1/Prime[k]; f[x_, 0] = x; z = 10;
%t A270487 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270487 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270487 x = (1/2)^(1/3); Table[n[x, k], {k, 1, z}]
%o A270487 (PARI) r(k) = 1/prime(k);
%o A270487 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270487 a(k, x=(1/2)^(1/3)) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 31 2016
%Y A270487 Cf. A269993, A000040.
%K A270487 nonn,frac,easy
%O A270487 1,2
%A A270487 _Clark Kimberling_, Mar 30 2016