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A270001 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1,1/2,1/3,1/4,...)

%I A270001 #13 Feb 16 2025 08:33:30
%S A270001 3,15,275,382677,1046649251798,2422932913436254796909358,
%T A270001 7298956212857760367589586285406004970615840077289,
%U A270001 146254918268677439622519920044753993861333975570554456887622664610038423045957554361330873143236585
%N A270001 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1,1/2,1/3,1/4,...)
%C A270001 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 A270001 See A269993 for a guide to related sequences.
%H A270001 Clark Kimberling, <a href="/A270001/b270001.txt">Table of n, a(n) for n = 1..12</a>
%H A270001 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270001 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270001 1/e = 1/3 + 1/(2*15) + 1/(3*275) + ...
%t A270001 r[k_] := 1/k; f[x_, 0] = x; z = 10;
%t A270001 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270001 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270001 x = 1/E; Table[n[x, k], {k, 1, z}]
%Y A270001 Cf. A269993.
%K A270001 nonn,frac,easy
%O A270001 1,1
%A A270001 _Clark Kimberling_, Mar 15 2016