Champernowne Constant

Champernowne's constant

(1)

(OEIS A033307) is the number obtained by concatenating the positive integers and interpreting them as decimal digits to the right of a decimal point. It is normal in base 10 (Champernowne 1933, Bailey and Crandall 2002). Mahler (1937) showed it to also be transcendental. The constant has been computed to digits by E. W. Weisstein (Jul. 3, 2013) using the Wolfram Language.

The infinite sequence of decimal digits in Champernowne's constant is sometimes known as Barbier's infinite word (Allouche and Shallit 2003, pp. 114, 299 and 334).

The numbers of decimal digits in the concatenations of the first primes, for , 2, ..., are 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, ... (OEIS A068670).

The Champernowne constant continued fraction contains sporadic very large terms, making the continued fraction difficult to calculate. However, the sizes of the continued fraction high-water marks display apparent patterns (Sikora 2012). The Copeland-Erdős constant, obtained by concatenating the primes (instead of all the positive integers), shows no comparable large-term behavior in the illustrated initial terms of its continued fraction.

The base- Champernowne constant is implemented in the Wolfram Language as ChampernowneNumber[b]. The base-2 and base-3 Champernowne constants are known as the binary Champernowne constant and ternary Champernowne constant, respectively.

For an integer base , a nested sum for the -ary Champernowne constant is given by

(2)

where is the floor function.

An explicit formula for the -ary Champernowne constant is given by

(3)

where is an integer and

			(4)
			(5)

(S. T. Parkin, pers. comm.). Writing , , and , the analytic expression for the addend is

(6)

It follows by summing the finite geometric-arithmetic series . This allows partial sums of the block expansion to be computed directly from the base without explicitly concatenating its digits.