A270380 Denominators of r-Egyptian fraction expansion for log(2), where r = (1,1/4,1/9,1/16,...).

2, 2, 2, 5, 437, 302513, 105578997456, 43374115871914236473230, 3061387825626545156370895519674610145110416238, 7629373903475555167995753649986186306421860702154239431571840759012861049320099293062744031

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

log(2) = 1/2 + 1/(4*2) + 1/(9*2) + 1/(16*5) + 1/(25*437) + ...

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 = Log(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=log(2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

Crossrefs

Cf. A269993.

Sequence in context: A208054 A059814 A079443 * A241460 A167568 A361301

Adjacent sequences: A270377 A270378 A270379 * A270381 A270382 A270383

Keyword

nonn,frac,easy,changed

Author

Clark Kimberling, Mar 20 2016

Status

approved