A270582 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/(k+1).

2, 3, 5, 65, 6529, 136091233, 41625259047416909, 2189507051227161558033650829868135, 75931290362065676573711484986356332365619562746656079489987281066955

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/(4*5) + 1/(5*65) + ...

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

Crossrefs

Cf. A269993.

Sequence in context: A259382 A103110 A042239 * A270916 A029975 A042787

Adjacent sequences: A270579 A270580 A270581 * A270583 A270584 A270585

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 03 2016

Status

approved