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A117116 Denominators of an Egyptian Fraction for phi = (1+sqrt(5))/2.

%I A117116 #40 Mar 19 2025 07:41:47
%S A117116 1,2,9,145,37986,2345721887,26943815937041299094,
%T A117116 811625643619814151937413504618770581764,
%U A117116 697120590223140234675813998970770820981012350673738243594006422610850113672220
%N A117116 Denominators of an Egyptian Fraction for phi = (1+sqrt(5))/2.
%C A117116 For each term, the largest possible unit fraction is used.
%H A117116 Amiram Eldar, <a href="/A117116/b117116.txt">Table of n, a(n) for n = 0..11</a>
%H A117116 Mohammad K. Azarian, <a href="http://www.jstor.org/stable/10.4169/college.math.j.42.4.329">Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958</a>, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. <a href="http://www.jstor.org/stable/10.4169/college.math.j.43.4.337">Solution</a> published in Vol. 43, No. 4, September 2012, pp. 340-342.
%H A117116 David Eppstein, <a href="http://www.ics.uci.edu/%7Eeppstein/numth/egypt/intro.html">Algorithms for Egyptian Fractions</a>.
%H A117116 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>.
%H A117116 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>.
%e A117116 a(4) = 145 because 1/145 is the largest unit fraction less than phi - 1/1 - 1/2 - 1/9.
%p A117116 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
%t A117116 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 *)
%o A117116 (PARI) x = (1 + sqrt(5))/2 - 1;
%o A117116 f(x, k) = if(k<1, x, f(x, k - 1) - 1/n(x, k));
%o A117116 n(x, k) = ceil(1/f(x, k - 1));
%o A117116 for(k = 0, 9, print1(if(k==0, 1, n(x, k)), ", ")) \\ _Indranil Ghosh_, Mar 27 2017
%Y A117116 Cf. A001622.
%K A117116 nonn,frac
%O A117116 0,2
%A A117116 _Paolo P. Lava_ and _Giorgio Balzarotti_, Apr 19 2006
%E A117116 Edited by _Don Reble_, Apr 21 2006