A006487 Denominators of greedy Egyptian fraction for square root of 2. (Formerly M2962)

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

Offset

0,2

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

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

Formula

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

Example

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

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 *)

Crossrefs

Sequence in context: A111431 A015701 A220294 * A240618 A042823 A132560

Adjacent sequences: A006484 A006485 A006486 * A006488 A006489 A006490

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

a(8) from Manfred Scheucher, Aug 17 2015

Status

approved