A270477 - OEIS

%I #8 Feb 24 2018 11:53:07

%S 1,5,19,909,709338,4794024440479,18787437394610419733587349,

%T 438597049892989902759955952867127541411726874886473,

%U 175915259950103387380668466916070098283235189077796884344520632101017268238077131833609385455236441012

%N Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r(k) = 1/Prime(k).

%C 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 See A269993 for a guide to related sequences.

%H Clark Kimberling, <a href="/A270477/b270477.txt">Table of n, a(n) for n = 1..11</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%e sqrt(1/3) = 1/(2*1) + 1/(3*5) + 1/(5*19) + 1/(7*909) + ...

%t r[k_] := 1/Prime[k]; f[x_, 0] = x; z = 10;

%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]

%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]

%t x = Sqrt[1/3]; Table[n[x, k], {k, 1, z}]

%Y Cf. A269993, A000040.

%K nonn,frac,easy

%O 1,2

%A _Clark Kimberling_, Mar 30 2016