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).