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A270591 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/(k+1). 2

%I #16 Dec 12 2023 18:51:50

%S 1,2,2,99,12204,249462465,93524017020207705,

%T 8528549813750403709101762452858246,

%U 70071914165301390868341700110703069865385640933927590404095892463912

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

%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="/A270591/b270591.txt">Table of n, a(n) for n = 1..12</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/(2*1) + 1/(3*2) + 1/(4*2) + 1/(5*99) + ...

%t r[k_] := 1/(k+1); 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.

%K nonn,frac,easy

%O 1,2

%A _Clark Kimberling_, Apr 04 2016

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Last modified May 28 01:34 EDT 2024. Contains 372900 sequences. (Running on oeis4.)