Champernowne constant

Yesterday, on the plane from Jakarta to Singapore, I was browsing The Story of Numbers by Mallik Asok Kumar and came across the following entry shown in Figure 1:

Figure 1: extract from page 114 of Mallik's The Story of Numbers

This struck a chord because earlier in the day, sitting in Starbuck's at Jakarta airport and analysing my number of the day (25554), I'd come across this entry in the Online Encyclopaedia of Integer Sequences (OEIS) shown in Figure 2:

Figure 2: OEIS A224896

What this meant was the sequence of digits 666666 occurs at position number 25554 in the decimal expansion of the Champernowne constant. Today, I had a look at the information contained in the Wikipedia entry for this topic. It explained that the Champernowne constant can be expressed as an infinite sequence:$$C_{m}=\sum_{n=1}^\infty\frac n{10_b^{~\left(\sum\limits_{k=1}^n\left\lceil\log_{10_b}(k+1)\right\rceil\right)}}$$where \(\lceil{x}\rceil = \)ceiling(\(x\)), \(10_b^{~x}=b^x\) in base 10, \(\log_{10_b}(x)=\log_{b_{10}}(x)\) and \(b\) is the base of the constant. However, a slightly different expression is given by Eric W. Weisstein (MathWorld):$$C_{m}=\sum_{n=1}^\infty\frac n{m^{\left(n+\sum\limits_{k=1}^{n-1}\left\lfloor\log_m(k+1)\right\rfloor\right)}}$$where \(\lfloor{x}\rfloor =\) floor function. I've no idea how these summations were arrived at but in order to determine positions listed in Figure 2, the SageMath code shown in Figure 4 can be used. The example shows the position of 88888888:

Figure 4: screenshot showing position of 88888888.
Click here to verify yourself in SageMathCell

Furthermore, the general name given to sequences like OEIS A224896 is the Earls Sequence, as explained in Figure 3.

Figure 3: The Earls Sequence showing the Champernowne constant
as well as other well known constants

Getting back to Champernowne's constant, its continued fraction expansion does not terminate (because it is not rational) and is aperiodic (because it is not an irreducible quadratic, Kurt Mahler having shown that the constant is transcendental). The terms in the continued fraction expansion exhibit very erratic behaviour, with extremely large terms appearing between many small ones e.g.

[0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15,

4 57540 11139 10310 76483 64662 82429 56118 59960 39397 10457 55500 06620 04393 09026 26592 56314 93795 32077 47128 65631 38641 20937 55035 52094 60718 30899 84575 80146 98631 48833 59214 17830 10987,

6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54, ... ]

To quote further from Wikipedia:

The large number at position 19 has 166 digits, and the next very large term at position 41 of the continued fraction has 2504 digits. The fact that there are such large numbers as terms of the continued fraction expansion is equivalent to saying that the convergents obtained by stopping before these large numbers provide an exceptionally good approximation of the Champernowne constant.