A270588 Denominators of r-Egyptian fraction expansion for e - 2, where r(k) = 1/(k+1).

1, 2, 5, 124, 73794, 5284326699, 229125959519570751554, 95635936955933623982265743051662580385628, 16608415739298461036456146606021108793638909234932647123655496824423388126111599755

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

e - 2 = 1/(2*1) + 1/(3*2) + 1/(4*5) + 1/(5*124) + ...

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

Crossrefs

Cf. A269993.

Sequence in context: A276109 A270481 A225944 * A214648 A175978 A092922

Adjacent sequences: A270585 A270586 A270587 * A270589 A270590 A270591

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 03 2016

Status

approved