Champernowne Constant Digits
The Champernowne constant has decimal expansion
(OEIS A033307).
The Earls sequence (starting position of 
copies of the digit 
) for 
is given for 
, 2, ... by 1, 34, 56, 1222, 1555, 25554, 29998, 433330, 7988888882, 1101010101010, ... (OEIS A224896).
The starting positions of the first occurrence of 
, 1, 2, ... in the decimal expansion of 
(not including the initial 0 to the left of the decimal point) are 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, ... (OEIS A229186).
Scanning the decimal expansion of 
until all 
-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 0, 00, 000, 0000, ..., which end at digits 11, 192, 2893, 38894, 488895, ... (OEIS A072290).
The digit sequence 0123456789 first occurs at positions 11234567799, 22345677908, 33456779017, 44567790126, 55677901235, 66779012344, ... (OEIS A000000) and 9876543210 at positions 7777777779, 9876543212, 19987654323, 30998765434, 42099876545, 53209987656, 64320998767, ... (OEIS A000000; E. Weisstein, Jul. 26, 2013).
-constant primes occur for 10, 14, 24, 235, 2804, 4347, 37735, ... (OEIS A071620) digits.
It is known that the Champernowne constant is normal in base 10 (Champernowne 1933, Bailey and Crandall 2002), though the following table giving the counts of digits in the first 
terms shows non-normal behavior up to at least 
due to an excess of 1s and surfeit of 0s when cutting the digit string off at locations such as 
.
 |
OEIS |
10 |
100 |
 |
 |
 |
 |
 |
 |
 |
| 0 |
A000000 |
0 |
5 |
66 |
747 |
8642 |
83528 |
884151 |
9234568 |
96021948 |
| 1 |
A000000 |
2 |
16 |
177 |
1858 |
19753 |
179810 |
1582562 |
14234568 |
130589850 |
| 2 |
A000000 |
1 |
16 |
177 |
1636 |
11111 |
94539 |
995260 |
10345679 |
100589849 |
| 3 |
A000000 |
1 |
16 |
148 |
858 |
8642 |
94539 |
995161 |
10234568 |
96589849 |
| 4 |
A000000 |
1 |
16 |
77 |
858 |
8642 |
94539 |
995160 |
9345679 |
96089849 |
| 5 |
A000000 |
1 |
11 |
77 |
858 |
8642 |
93723 |
982462 |
9345679 |
96029849 |
| 6 |
A000000 |
1 |
5 |
77 |
858 |
8642 |
93538 |
895160 |
9345679 |
96022849 |
| 7 |
A000000 |
1 |
5 |
67 |
833 |
8642 |
93538 |
894462 |
9345679 |
96022049 |
| 8 |
A000000 |
1 |
5 |
67 |
747 |
8642 |
88718 |
891462 |
9333333 |
96021959 |
| 9 |
A000000 |
1 |
5 |
67 |
747 |
8642 |
83528 |
884160 |
9234568 |
96021949 |
REFERENCES:
Champernowne, D. G. "The Construction of Decimals Normal in the Scale of Ten." J. London Math. Soc. 8, 1933.
Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.
Sloane, N. J. A. Sequences A071620, A072290, A224896, and A229186 in "The On-Line Encyclopedia of Integer Sequences."
