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A270525
Denominators of r-Egyptian fraction expansion for log(2), where r(k) = 1/k!.
1
2, 3, 7, 16, 125, 8861, 22953346, 75396230717172, 12599316539774886331286770578, 16091945901447902466261129483788815818314990863881046593, 148472040746053348386996212070544963774172473939368496909380855505454576714141628510687134372727919512688601726
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.
EXAMPLE
log(2) = 1/(1*2) + 1/(2*3) + 1/(6*7) + 1/(24*16) + ...
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 = Log[2]; Table[n[x, k], {k, 1, z}]
PROG
(PARI) r(k) = 1/k!;
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 31 2016
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Clark Kimberling, Mar 30 2016
STATUS
approved