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A270347 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/2,1/4,1/8,...) 3
2, 3, 7, 27, 650, 689392, 1130869248534, 2046949388776880512222550, 5664769376602746621028306587399157369622446276283, 61600875764518391286867927949695082949269716944423018977948114995142883041085134431474743108010213 (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..20

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

EXAMPLE

sqrt(1/2) = 1/2 + 1/(2*3) + 1/(4*7) + ...

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 = Sqrt[1/2]; 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=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 18 2016

CROSSREFS

Cf. A269993.

Sequence in context: A156142 A052877 A137075 * A060412 A276665 A062573

Adjacent sequences:  A270344 A270345 A270346 * A270348 A270349 A270350

KEYWORD

nonn,frac,easy

AUTHOR

Clark Kimberling, Mar 17 2016

STATUS

approved

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Last modified September 21 15:22 EDT 2017. Contains 292300 sequences.