A270487 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/Prime(k).

1, 2, 2, 6, 29, 860, 626907, 1582796431872, 4577382865450526674426008, 77218331531088831524423800072197013265311322482652, 10410509369911993512345323774444196964795747018426948027297775848734862056109801420845614477793011811

Offset

1,2

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.

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

Example

(1/2)^(1/3) = 1/(2*1) + 1/(3*2) + 1/(5*2) + 1/(7*6) + ...

Mathematica

r[k_] := 1/Prime[k]; 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}]

Prog

(PARI) r(k) = 1/prime(k);
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 31 2016

Crossrefs

Cf. A269993, A000040.

Sequence in context: A179320 A004304 A326907 * A058250 A179929 A278258

Adjacent sequences: A270484 A270485 A270486 * A270488 A270489 A270490

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 30 2016

Status

approved