A270589 Denominators of r-Egyptian fraction expansion for log(2), where r(k) = 1/(k+1).

1, 2, 10, 136, 16792, 481484601, 5752738775151622145, 32677165184182243545746422896718822871, 3003270816051953126782801862429770277283389802341096352218029665206283079575

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..12

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

log(2) = 1/(2*1) + 1/(3*2) + 1/(4*10) + 1/(5*136) + ...

Mathematica

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

Crossrefs

Cf. A269993.

Sequence in context: A075199 A134981 A087417 * A171214 A213955 A091990

Adjacent sequences: A270586 A270587 A270588 * A270590 A270591 A270592

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 04 2016

Status

approved