|
|
A270316
|
|
Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1,1/2,1/3,1/4,...)
|
|
2
|
|
|
2, 2, 8, 123, 149367, 19877572990, 3398650153657920854371, 38501744904404393452660892011327652171148221, 1751742507912624184333715455628345093210972368514121272905550101268413741408122585972087
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
EXAMPLE
|
(1/2)^(1/3) = 1/2 + 1/(2*2) + 1/(3*8) + ...
|
|
MATHEMATICA
|
r[k_] := 1/k; 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 = (1/2)^(1/3); Table[n[x, k], {k, 1, z}]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,frac,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|