A117116 Denominators of an Egyptian Fraction for phi = (1+sqrt(5))/2.
1, 2, 9, 145, 37986, 2345721887, 26943815937041299094, 811625643619814151937413504618770581764, 697120590223140234675813998970770820981012350673738243594006422610850113672220
Offset: 0
Examples
a(4) = 145 because 1/145 is the largest unit fraction less than phi - 1/1 - 1/2 - 1/9.
Links
- Amiram Eldar, Table of n, a(n) for n = 0..11
- Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342.
- David Eppstein, Algorithms for Egyptian Fractions.
- Eric Weisstein's World of Mathematics, Egyptian Fraction.
- Index entries for sequences related to Egyptian fractions.
Crossrefs
Cf. A001622.
Programs
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Maple
v[0]:=1: for n from 1 to 10 do x:=ceil(1/((1+sqrt(5))/2-add(1/v[i],i=0..n-1)));while not x::integer do Digits:=2*Digits; x:=ceil(1/((1+sqrt(5))/2-add(1/v[i],i=0..n-1))) od; v[n]:=x; od: seq(v[i],i=0..8); # Paolo P. Lava, May 03 2018
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Mathematica
a = {1}; k = N[(Sqrt[5] - 1)/2, 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a (* Artur Jasinski, Sep 22 2008 *) -
PARI
x = (1 + sqrt(5))/2 - 1; f(x, k) = if(k<1, x, f(x, k - 1) - 1/n(x, k)); n(x, k) = ceil(1/f(x, k - 1)); for(k = 0, 9, print1(if(k==0, 1, n(x, k)), ", ")) \\ Indranil Ghosh, Mar 27 2017
Extensions
Edited by Don Reble, Apr 21 2006
Comments