Champernowne Constant

Champernowne's number 0.1234567891011... (Sloane's A033307) is the decimal obtained by concatenating the Positive Integers. It is Normal in base 10. In 1961, Mahler showed it to also be Transcendental.

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,

45754011139103107648364662824295611859960393971045755500066200439309026265925631493795320774712865631386412093755035520946071830899845758014698631488335921417830109876, 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, 247, 358, 460, ... and have 6, 166, 2504, 140, 33102, 109, 2468, 136, ... digits (Plouffe). Interestingly, the Copeland-Erdös Constant, which is the decimal obtained by concatenating the Primes, has a well-behaved Continued Fraction which does not show the ``large term'' phenomenon.

See also Copeland-Erdös Constant, Smarandache Sequences

References

Champernowne, D. G. ``The Construction of Decimals Normal in the Scale of Ten.'' J. London Math. Soc. 8, 1933.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html

Sloane, N. J. A. A030167 and A033307 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.