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A270518 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r(k) = 1/k!. 1
2, 7, 29, 239, 35642, 4939700112, 48108453420633293272, 444429875521548685791697227054499321900, 25562938514216590071082104331351977875333056562865491878765431482309855946304 (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

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(1/3) = 1/(1*2) + 1/(2*7) + 1/(6*29) + 1/(24*239) + ...

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 = Sqrt[1/3]; 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=sqrt(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016

CROSSREFS

Cf. A269993, A000142, A020760.

Sequence in context: A143883 A003437 A192410 * A094475 A093034 A125174

Adjacent sequences:  A270515 A270516 A270517 * A270519 A270520 A270521

KEYWORD

nonn,frac,easy

AUTHOR

Clark Kimberling, Mar 30 2016

STATUS

approved

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Last modified April 19 08:34 EDT 2019. Contains 322241 sequences. (Running on oeis4.)