A270581 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r(k) = 1/(k+1).

1, 5, 24, 750, 619143, 429785415881, 249018030532843762406536, 786190926449579590882811037530136188173125199986, 4363647658444437374958875225038381920365014054256819920344934200940398256657209564018610271709274

Offset

1,2

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) + 1/(3*5) + 1/(4*24) + 1/(5*750) + ...

Mathematica

r[k_] := 1/(k+1); 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}]

Crossrefs

Cf. A269993.

Sequence in context: A267279 A257214 A003224 * A156310 A142725 A270590

Adjacent sequences: A270578 A270579 A270580 * A270582 A270583 A270584

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 03 2016

Status

approved