A270478 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/Prime(k).

2, 3, 4, 47, 1445, 3111965, 60437225141058, 19833308022477607066005214665, 466985874016778023693751912505337681207396530069379830856, 214712731506707254615377967955272660569584599006507424981466453878259117882233362841865583894851904770121359232415

Offset

1,1

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..12

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

sqrt(2) - 1 = 1/(2*2) + 1/(3*3) + 1/(5*4) + 1/(7*47) + ...

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 = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]

Crossrefs

Cf. A269993, A000040.

Sequence in context: A037322 A037429 A230452 * A235495 A257482 A023167

Adjacent sequences: A270475 A270476 A270477 * A270479 A270480 A270481

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 30 2016

Status

approved