A270379 Denominators of r-Egyptian fraction expansion for e - 2, where r = (1,1/4,1/9,1/16,...).

2, 2, 2, 2, 7, 37, 1817, 3361666, 24283670558553, 1002770956493811911694552768, 843337841302004296404319706946194895734287215696998151, 890614335579920230119707369559263943501588363957602897846451247124061017888881480680329044310526970935480485

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..13

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

e - 2 = 1/2 + 1/(2*2) + 1/(9*2) + 1/(16*2) + 1/(25*7) +  ...

Mathematica

r[k_] := 1/k^2; 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}]

Prog

(PARI) r(k) = 1/k^2;
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=exp(1)-2) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

Crossrefs

Cf. A269993.

Sequence in context: A138068 A246052 A054083 * A337358 A295557 A327641

Adjacent sequences: A270376 A270377 A270378 * A270380 A270381 A270382

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 20 2016

Status

approved