A270480 Denominators of r-Egyptian fraction expansion for (golden ratio - 1), where r(k) = 1/Prime(k).

1, 3, 29, 5427, 36287610, 1365965619077845, 23235868400278912260438706402888, 522919219334412314763983041497942273235588152390441601515859340

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..11

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

tau - 1 = 1/(2*1) + 1/(3*3) + 1/(5*29) + 1/(7*5427) + ...

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 = GoldenRatio - 1; Table[n[x, k], {k, 1, z}]

Crossrefs

Cf. A269993, A000040.

Sequence in context: A322451 A213792 A133663 * A229051 A006526 A139517

Adjacent sequences: A270477 A270478 A270479 * A270481 A270482 A270483

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 30 2016

Status

approved