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

Author

Simon Plouffe

Keywords

• nonn
• cons

Comments

Kempner shows that numbers of a general form (which includes this constant) are transcendental. - Charles R Greathouse IV, Nov 07 2017

Examples

			0.81642150902189314370....
		

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.
• Index entries for transcendental numbers

Crossrefs

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

Programs

Mathematica

RealDigits[ N[ Sum[1/2^(2^n), {n, 0, Infinity}], 110]] [[1]]

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

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

Extensions

Edited by Robert G. Wilson v, Dec 11 2002

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