A270382 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1,1/4,1/9,1/16,...).

2, 1, 3, 10, 97, 24851, 510157381, 695243618523592916, 2521217027896573870788274798987969315, 200759268273854851798439056384882383919258596635924900200845873520031055851

Offset: 1

Author

Clark Kimberling, Mar 22 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.

See A269993 for a guide to related sequences.

Examples

			(1/2)^(1/3) = 1/2 + 1/(4*1) + 1/(9*3) + 1/(16*10) + ...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..13
• Eric Weisstein's World of Mathematics, Egyptian Fraction
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A269993, A270714.

Programs

Mathematica

r[k_] := 1/k^2; 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 = (1/2)^(1/3); Table[n[x, k], {k, 1, z}]

PARI

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=(1/2)^(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016