cp's OEIS Frontend

A270347 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/2,1/4,1/8,...)

%I A270347 #14 Feb 16 2025 08:33:31
%S A270347 2,3,7,27,650,689392,1130869248534,2046949388776880512222550,
%T A270347 5664769376602746621028306587399157369622446276283,
%U A270347 61600875764518391286867927949695082949269716944423018977948114995142883041085134431474743108010213
%N A270347 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/2,1/4,1/8,...)
%C A270347 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 A270347 See A269993 for a guide to related sequences.
%H A270347 Clark Kimberling, <a href="/A270347/b270347.txt">Table of n, a(n) for n = 1..12</a>
%H A270347 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270347 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270347 sqrt(1/2) = 1/2 + 1/(2*3) + 1/(4*7) + ...
%t A270347 r[k_] := 2/2^k; f[x_, 0] = x; z = 10;
%t A270347 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270347 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270347 x = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]
%o A270347 (PARI) r(k) = 2/2^k;
%o A270347 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270347 a(k, x=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 18 2016
%Y A270347 Cf. A269993.
%K A270347 nonn,frac,easy
%O A270347 1,1
%A A270347 _Clark Kimberling_, Mar 17 2016