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

1, 3, 13, 253, 218201, 61323543802, 5704059172637470075854, 178059816815203395552917056787722451335939040, 227569456678536847041583520060628448125647436561262746582115170178319521793841532532509636

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Comments

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

Examples

			sqrt(2) = 1 + 1/3 + 1/13 + 1/253 + 1/218201 + ... . - _Jon E. Schoenfield_, Dec 26 2014
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Manfred Scheucher, Table of n, a(n) for n = 0..11
• Manfred Scheucher, Table of n, a(n) for n = 0..14
• Manfred Scheucher, Sage Script.
• D. S. Kluk and N. J. A. Sloane, Correspondence, 1979.
• Eric Weisstein's World of Mathematics, Egyptian Fraction.
• Index entries for sequences related to Egyptian fractions

Programs

Maple

a[0]:= 1;
for n from 1 to 10 do
  v:= ceil(1/(sqrt(2)-add(1/a[i],i=0..n-1)));
  while not v::integer do
    Digits:= 2*Digits;
    v:= ceil(1/(sqrt(2)-add(1/a[i],i=0..n-1)))
  od;
  a[n]:= v;
od:
seq(a[i],i=0..10); # Robert Israel, Aug 17 2015

Mathematica

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 *)

Formula

a(n) = ceiling(1/(sqrt(2) - Sum_{j=0..n-1} 1/a(j))). - Jon E. Schoenfield, Dec 26 2014

Extensions

a(8) from Manfred Scheucher, Aug 17 2015