The name Smarandache number is used here for a term of the consecutive number sequences formed by concatenating the decimal representations of the first
positive integers, where 
: 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). For 
, the terms are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
These are sometimes called Smarandache consecutive numbers. Similarly, a Smarandache number that is prime will be called a Smarandache prime. The Great Smarandache PRPrime search reported no prime terms through index 344869 as of 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 
and 
is the floor function.
 |  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 positive 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 of 
where 
are the digits of 
gives a plot showing batrachion-like
structure (left figure), and first omitting the leading 1 from each concatenated
binary integer and then taking the same cumulative
sum (right figure) gives structures resembling the blancmange
function (and the Hofstadter-Conway
$10,000 sequence).
