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A270354 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1, 1/2, 1/4, 1/8, ...)

%I A270354 #16 Feb 16 2025 08:33:31
%S A270354 3,15,207,24777,1797835772,2401072239422894903,
%T A270354 36947191921380265723491992928675837908,
%U A270354 1242004943621920150072266455052958650167034792376067355585774287542963919184
%N A270354 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1, 1/2, 1/4, 1/8, ...)
%C A270354 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 A270354 See A269993 for a guide to related sequences.
%H A270354 Clark Kimberling, <a href="/A270354/b270354.txt">Table of n, a(n) for n = 1..11</a>
%H A270354 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270354 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270354 1/e = 1/3 + 1/(2*15) + 1/(4*207) + ...
%t A270354 r[k_] := 2/2^k; f[x_, 0] = x; z = 10;
%t A270354 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270354 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270354 x = 1/E; Table[n[x, k], {k, 1, z}]
%o A270354 (PARI) r(k) = 2/2^k;
%o A270354 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270354 a(k, x=exp(-1)) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 18 2016
%Y A270354 Cf. A269993.
%K A270354 nonn,frac,easy
%O A270354 1,1
%A A270354 _Clark Kimberling_, Mar 17 2016