

A270348


Denominators of rEgyptian fraction expansion for sqrt(1/3), where r = (1,1/2,1/4,1/8,...)


1



2, 7, 43, 1161, 796510, 1101781866330, 648667164391834988511313, 521313118065995695198529265268104396429334449023, 177042477384698216444912803612486097958997328262217304760270340328784709181787835108737458616981
(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(k1)), and f(k) = f(k1)  r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the rEgyptian 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

sqrt(1/3) = 1/2 + 1/(2*7) + 1/(4*43) + ...


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[1/3]; Table[n[x, k], {k, 1, z}]


PROG

(PARI) r(k) = 2/2^k;
f(k, x) = if (k==0, x, f(k1, x)  r(k)/a(k, x); );
a(k, x=sqrt(1/3)) = ceil(r(k)/f(k1, x)); \\ Michel Marcus, Mar 18 2016


CROSSREFS

Cf. A269993.
Sequence in context: A011835 A198946 A212270 * A270580 A103084 A041507
Adjacent sequences: A270345 A270346 A270347 * A270349 A270350 A270351


KEYWORD

nonn,frac,easy


AUTHOR

Clark Kimberling, Mar 17 2016


STATUS

approved



