A269573 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1,1,1,1,1,...)

2, 4, 23, 4500, 23314202, 703143261541584, 580028504455491926110281336263, 471554575224119231041268294704259548817134505334232514876247

Offset: 1

Author

Clark Kimberling, Mar 15 2016

Keywords

• nonn
• frac
• easy

Comments

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.

Examples

			(1/2)^(1/3) = 1/2 + 1/4 + 1/23 + ...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..11
• Eric Weisstein's World of Mathematics, Egyptian Fraction
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A269993 (guide to related sequences).

Programs

Mathematica

r[k_] := 1; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = 2^(-1/3); Table[n[x, k], {k, 1, z}]  (* A269573 *)