OFFSET
0,1
COMMENTS
Kempner shows that numbers of a general form (which includes this constant) are transcendental. - Charles R Greathouse IV, Nov 07 2017
REFERENCES
M. J. Knight, An "oceans of zeros" proof that a certain non-Liouville number is transcendental, The American Mathematical Monthly, Vol. 98, No. 10 (1991), pp. 947-949.
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..20000
Boris Adamczewski, The Many Faces of the Kempner Number, Journal of Integer Sequences, Vol. 16 (2013), #13.2.15.
David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl Pomerance, On the Binary Expansions of Algebraic Numbers, Journal de Théorie des Nombres de Bordeaux, volume 16, number 3, 2004, pages 487-518. Also LBNL-53854 2003, and authors' copies one, four.
D. H. Bailey and H. R. P. Ferguson, Numerical results on relations between fundamental constants using a new algorithm, Mathematics of Computation, Vol.53 No. 188 (1989), 649-656. (Annotated scanned copy)
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
Aubrey J. Kempner, On transcendental numbers, Transactions of the American Mathematical Society 17 (1916), pp. 476-482.
Simon Plouffe, Plouffe's Inverter, sum(1/2^(2^n), n=0..infinity); to 20000 digits
Simon Plouffe, sum(1/2^(2^n), n=0..infinity) to 1024 digits
Jeffrey Shallit, Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), no. 2, 209-217.
FORMULA
Equals -Sum_{k>=1} mu(2*k)/(2^k - 1) = Sum_{k>=1, k odd} mu(k)/(2^k - 1). - Amiram Eldar, Jun 22 2020
EXAMPLE
0.81642150902189314370....
MATHEMATICA
RealDigits[ N[ Sum[1/2^(2^n), {n, 0, Infinity}], 110]] [[1]]
PROG
(PARI) default(realprecision, 20080); x=suminf(n=0, 1/2^(2^n)); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b007404.txt", n, " ", d)); \\ Harry J. Smith, May 07 2009
(PARI) suminf(k = 0, 1/(2^(2^k))) \\ Michel Marcus, Mar 26 2017
(PARI) suminf(k=0, 1.>>2^k) \\ Charles R Greathouse IV, Nov 07 2017
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
EXTENSIONS
Edited by Robert G. Wilson v, Dec 11 2002
Deleted old PARI program Harry J. Smith, May 20 2009
STATUS
approved