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A269994 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1,1/2,1/3,1/4,...) 2
2, 7, 57, 3391, 10010183, 588972486242552, 961457184347597076119863109462, 2244227167765735741796211572067153905745156229769919746729015 (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) = 1/2 + 1/(2*7) + 1/(3*57) + ...

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;

x = sqrt(1/3);

f(x, k) = if(k<1, x, f(x, k - 1) - r(k)/n(x, k));

n(x, k) = ceil(r(k)/f(x, k - 1));

for(k = 1, 8, print1(n(x, k), ", ")) \\ Indranil Ghosh, Mar 27 2017, translated from Mathematica code

CROSSREFS

Cf. A269993.

Sequence in context: A121079 A270395 A105183 * A023364 A120952 A270404

Adjacent sequences:  A269991 A269992 A269993 * A269995 A269996 A269997

KEYWORD

nonn,frac,easy

AUTHOR

Clark Kimberling, Mar 15 2016

STATUS

approved

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Last modified January 21 10:26 EST 2020. Contains 331105 sequences. (Running on oeis4.)