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A270352 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1, 1/2, 1/4, 1/8, ...) 1
4, 8, 44, 977, 498723, 138012074956, 45087947486104434546449, 2223745971024423874814212532278502253766982404, 3439676840537267257806008796995789895364959784333600339427716437786254731225969490712842205 (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..11

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

EXAMPLE

1/Pi = 1/4 + 1/(2*8) + 1/(4*44) + ...

MATHEMATICA

r[k_] := 2/2^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/Pi; Table[n[x, k], {k, 1, z}]

PROG

(PARI) r(k) = 2/2^k;

f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );

a(k, x=1/Pi) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 18 2016

CROSSREFS

Cf. A269993.

Sequence in context: A285751 A189538 A351071 * A183394 A291948 A002470

Adjacent sequences:  A270349 A270350 A270351 * A270353 A270354 A270355

KEYWORD

nonn,frac,easy

AUTHOR

Clark Kimberling, Mar 17 2016

STATUS

approved

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Last modified May 24 17:57 EDT 2022. Contains 354043 sequences. (Running on oeis4.)