cp's OEIS Frontend

A270356 Denominators of r-Egyptian fraction expansion for log(2), where r = (1, 1/2, 1/4, 1/8, ...)

%I A270356 #16 Feb 16 2025 08:33:31
%S A270356 2,3,10,85,6297,105324757,10291333539500676,
%T A270356 72129634294824118806681649563665,
%U A270356 3614136206345221874912341551952565198060297016360952863886217259
%N A270356 Denominators of r-Egyptian fraction expansion for log(2), where r = (1, 1/2, 1/4, 1/8, ...)
%C A270356 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 A270356 See A269993 for a guide to related sequences.
%H A270356 Clark Kimberling, <a href="/A270356/b270356.txt">Table of n, a(n) for n = 1..12</a>
%H A270356 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270356 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270356 log(2) = 1/2 + 1/(2*3) + 1/(4*10) + ...
%t A270356 r[k_] := 2/2^k; f[x_, 0] = x; z = 10;
%t A270356 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270356 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270356 x = Log(2); Table[n[x, k], {k, 1, z}]
%o A270356 (PARI) r(k) = 2/2^k;
%o A270356 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270356 a(k, x=log(2)) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 18 2016
%Y A270356 Cf. A269993.
%K A270356 nonn,frac,easy
%O A270356 1,1
%A A270356 _Clark Kimberling_, Mar 17 2016