|
%I #12 Feb 23 2018 22:03:54
%S 2,3,7,63,7179,142233093,64600110035609517,
%T 5529148350206824361693538422450743,
%U 39876890198849678230595649918157265458164953427845442505533508344048
%N Denominators of r-Egyptian fraction expansion for e - 2, where r = (1,1/2,1/3,1/4,...)
%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="/A270002/b270002.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 e - 2 = 1/2 + 1/(2*3) + 1/(3*7) + ...
%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 = E - 2; Table[n[x, k], {k, 1, z}]
%Y Cf. A269993.
%K nonn,frac,easy
%O 1,1
%A _Clark Kimberling_, Mar 15 2016
|
