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A006487 Denominators of greedy Egyptian fraction for square root of 2.

%I A006487 M2962 #50 Feb 16 2025 08:32:30
%S A006487 1,3,13,253,218201,61323543802,5704059172637470075854,
%T A006487 178059816815203395552917056787722451335939040,
%U A006487 227569456678536847041583520060628448125647436561262746582115170178319521793841532532509636
%N A006487 Denominators of greedy Egyptian fraction for square root of 2.
%C A006487 Conjecture: Let a(n) = 2^2^(n + b(n)), then b(n) converges to a constant that is about 0.2163... - _Manfred Scheucher_, Aug 17 2015
%D A006487 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006487 Manfred Scheucher, <a href="/A006487/b006487.txt">Table of n, a(n) for n = 0..11</a>
%H A006487 Manfred Scheucher, <a href="/A006487/a006487_1.txt">Table of n, a(n) for n = 0..14</a>
%H A006487 Manfred Scheucher, <a href="/A006487/a006487.sage.txt">Sage Script</a>.
%H A006487 D. S. Kluk and N. J. A. Sloane, <a href="/A002050/a002050_3.pdf">Correspondence, 1979</a>.
%H A006487 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>.
%H A006487 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%F A006487 a(n) = ceiling(1/(sqrt(2) - Sum_{j=0..n-1} 1/a(j))). - _Jon E. Schoenfield_, Dec 26 2014
%e A006487 sqrt(2) = 1 + 1/3 + 1/13 + 1/253 + 1/218201 + ... . - _Jon E. Schoenfield_, Dec 26 2014
%p A006487 a[0]:= 1;
%p A006487 for n from 1 to 10 do
%p A006487   v:= ceil(1/(sqrt(2)-add(1/a[i],i=0..n-1)));
%p A006487   while not v::integer do
%p A006487     Digits:= 2*Digits;
%p A006487     v:= ceil(1/(sqrt(2)-add(1/a[i],i=0..n-1)))
%p A006487   od;
%p A006487   a[n]:= v;
%p A006487 od:
%p A006487 seq(a[i],i=0..10); # _Robert Israel_, Aug 17 2015
%t A006487 lst={};k=N[Sqrt[2],1000];Do[s=Ceiling[1/k];AppendTo[lst,s];k=k-1/s,{n,12}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 02 2009 *)
%K A006487 nonn
%O A006487 0,2
%A A006487 _N. J. A. Sloane_, _Simon Plouffe_
%E A006487 a(8) from _Manfred Scheucher_, Aug 17 2015