Champernowne constant
Eric W. Weisstein
eww6n at carina.astro.virginia.edu
Mon Apr 27 17:05:40 CEST 1998
Hello, all.
As most of you probably known, Champernowne's number 0.1234567891011... is
the decimal constant obtained by concatenating the positive integers. The
continued fraction of the Champernowne constant is [0, 8, 9, 1, 149083, 1,
1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15,
457540111391031076483646628242956118599603939\
710457555000662004393090262659256314937953207\
747128656313864120937550355209460718308998457\
5801469863148833592141783010987, 6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2,
1, 83, 1, 156, 4, 58, 8, 54, ... (Sloane's A030167). The next term of
the continued fraction is huge, having 2504 digits. In fact, the
coefficients eventually become unbounded, making the continued fraction
difficult to calculate for too many more terms.
Large terms greater than $10^5$ occur at positions 5, 19, 41, 102, 163,
\dots{} and have 6, 166, 2504, 140, 21106, ... digits. I can't get
Mathematica to go beyond the 21,106-digit term (and don't have the
motivation to write a multiprecision C or FORTRAN program), so my
challenge to all of you is to extend the lists of positions of the huge
terms and the number of digits in each of those terms.
Regards,
- -Eric
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* Eric W. Weisstein phone: (804) 924-4899 *
* U. Virginia Astron. Dept. FAX: (804) 924-3104 *
* P.O. Box 3818 e-mail: eww6n at virginia.edu *
* Charlottesville, VA 22903 WWW: http://www.astro.virginia.edu/~eww6n *
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* Accept the inevitable; there's nothing you can do about it anyway. -me *
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