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A269994 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1,1/2,1/3,1/4,...)

%I A269994 #17 Feb 16 2025 08:33:30
%S A269994 2,7,57,3391,10010183,588972486242552,961457184347597076119863109462,
%T A269994 2244227167765735741796211572067153905745156229769919746729015
%N A269994 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1,1/2,1/3,1/4,...)
%C A269994 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 A269994 See A269993 for a guide to related sequences.
%H A269994 Clark Kimberling, <a href="/A269994/b269994.txt">Table of n, a(n) for n = 1..12</a>
%H A269994 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A269994 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A269994 sqrt(1/3) = 1/2 + 1/(2*7) + 1/(3*57) + ...
%t A269994 r[k_] := 1/k; f[x_, 0] = x; z = 10;
%t A269994 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A269994 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A269994 x = Sqrt[1/3]; Table[n[x, k], {k, 1, z}]
%o A269994 (PARI) r(k) = 1/k;
%o A269994 x = sqrt(1/3);
%o A269994 f(x, k) = if(k<1, x, f(x, k - 1) - r(k)/n(x, k));
%o A269994 n(x, k) = ceil(r(k)/f(x, k - 1));
%o A269994 for(k = 1, 8, print1(n(x, k),", ")) \\ _Indranil Ghosh_, Mar 27 2017, translated from Mathematica code
%Y A269994 Cf. A269993.
%K A269994 nonn,frac,easy
%O A269994 1,1
%A A269994 _Clark Kimberling_, Mar 15 2016