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A270476 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/Prime(k).

%I A270476 #10 Feb 16 2025 08:33:31
%S A270476 1,2,5,325,164073,23835564403,509747166181000498873,
%T A270476 590605960011761211516665913403247265840072,
%U A270476 493340534610970903685535778248091335992630045997033895220604001625216391426083646793
%N A270476 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/Prime(k).
%C A270476 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 A270476 See A269993 for a guide to related sequences.
%H A270476 Clark Kimberling, <a href="/A270476/b270476.txt">Table of n, a(n) for n = 1..12</a>
%H A270476 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270476 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270476 sqrt(1/2) = 1/(2*1) + 1/(3*2) + 1/(5*5) + 1/(7*325) + ...
%t A270476 r[k_] := 1/Prime[k]; f[x_, 0] = x; z = 10;
%t A270476 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270476 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270476 x = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]
%Y A270476 Cf. A269993, A000040.
%K A270476 nonn,frac,easy
%O A270476 1,2
%A A270476 _Clark Kimberling_, Mar 30 2016