Consider the consecutive number sequences formed by the concatenation of the first 
positive integers: 1,
12, 123, 1234, ... (OEIS A007908; Smarandache
1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998; Wolfram
2002, p. 913).
This sequence gives the digits of the Champernowne
constant, and is sometimes also known as the Barbier infinite word (Allouche
and Shallit 2003, pp. 114, 299, and 336). The terms up to 
are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
These are sometimes called Smarandache consecutive numbers, but in this work, the terms in the sequence will be called simply Smarandache numbers. Similarly, a Smarandache
number that is prime will be called a Smarandache
prime. Surprisingly, no Smarandache primes 
exist for 
(Great Smarandache PRPrime search; Dec. 5,
2016).
The number of digits of 
can be computed by noticing the pattern in the following
table, where
 |
(3)
|
is the number of digits in 
.
 |  range | digits |
| 1 | 1-9 |  |
| 2 | 10-99 |  |
| 3 | 100-999 |  |
| 4 | 1000-9999 |  |
By induction, the number of digits 
in 
can be written
 |  |  |
(4)
|
 |  |  |
(5)
|
where the second term is the repunit 
. For 
, 2, ..., the digit lengths 
of 
are therefore 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17,
19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, ... (OEIS A058183).
The results of concatenating the binary representations of the first few integers are 1, 110, 11011, 11011100, 11011100101, ... (OEIS A058935).
These digit sequences are plotted above for 
to 90. Interpreting the digit sequence as a binary fraction,
the result is the binary Champernowne
constant 
.
Interestingly, taking the cumulative sum 
where 
are the digits 
gives a plot showing batrachion-like
structure (left figure), and doing the same with 
(right figure) gives structures resembling
the Blancmange function (and the Hofstadter-Conway
$10,000 sequence).
