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A270517 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/k!. 2
2, 3, 5, 6, 52, 668, 171510, 4590170768, 17103459833953822083, 31906466290986600582512428032058109695, 237271596693541800921324673318278335675822001026279366213434934428597656224 (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..13

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

EXAMPLE

sqrt(1/2) = 1/(1*2) + 1/(2*3) + 1/(6*5) + 1/(24*6) + ...

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[1/2]; 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=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016

CROSSREFS

Cf. A269993, A000142, A010503.

Sequence in context: A263650 A076384 A261579 * A124648 A093339 A113566

Adjacent sequences:  A270514 A270515 A270516 * A270518 A270519 A270520

KEYWORD

nonn,frac,easy

AUTHOR

Clark Kimberling, Mar 30 2016

STATUS

approved

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Last modified January 23 19:12 EST 2020. Contains 331175 sequences. (Running on oeis4.)