A270583 - OEIS

%I #9 Feb 16 2025 08:33:31

%S 1,2,4,70,6174,45785878,9941815425565254,

%T 90769288470114014438337290256582,

%U 35712953028973795679646888004332118441804994642382862330509857037

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

%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="/A270583/b270583.txt">Table of n, a(n) for n = 1..12</a>

%H Eric Weisstein's World of Mathematics, <a href="https://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(3) - 1 = 1/(2*1) + 1/(3*2) + 1/(4*4) + 1/(5*70) + ...

%t r[k_] := 1/(k+1); 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[3] - 1; Table[n[x, k], {k, 1, z}]

%Y Cf. A269993.

%K nonn,frac,easy,changed

%O 1,2

%A _Clark Kimberling_, Apr 03 2016