A007404 Decimal expansion of Sum_{n>=0} 1/2^(2^n).

8, 1, 6, 4, 2, 1, 5, 0, 9, 0, 2, 1, 8, 9, 3, 1, 4, 3, 7, 0, 8, 0, 7, 9, 7, 3, 7, 5, 3, 0, 5, 2, 5, 2, 2, 1, 7, 0, 3, 3, 1, 1, 3, 7, 5, 9, 2, 0, 5, 5, 2, 8, 0, 4, 3, 4, 1, 2, 1, 0, 9, 0, 3, 8, 4, 3, 0, 5, 5, 6, 1, 4, 1, 9, 4, 5, 5, 5, 3, 0, 0, 0, 6, 0, 4, 8, 5, 3, 1, 3, 2, 4, 8, 3, 9, 7, 2, 6, 5, 6, 1, 7, 5, 5, 8

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, 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.

Index entries for transcendental numbers

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

Cf. A007400, A078885, A078585, A078886, A078887, A078888, A078889, A078890, A036987.

Sequence in context: A351211 A033812 A019717 * A299627 A157697 A281785

Adjacent sequences: A007401 A007402 A007403 * A007405 A007406 A007407

Keyword

nonn,cons

Author

Simon Plouffe

Extensions

Edited by Robert G. Wilson v, Dec 11 2002

Deleted old PARI program Harry J. Smith, May 20 2009

Status

approved