Consecutive number sequences are sequences constructed by concatenating numbers of a given type. Many of these sequences were considered by Smarandache, so they are sometimes known as Smarandache sequences.

The nth term of the consecutive integer sequence consists of the concatenation of the first n positive integers: 1, 12, 123, 1234, ... (Sloane's 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. The terms up to n = 9 are given by

Fleuren (1999) verified that the absence of primes up to n = 200, a result extended to the first terms by Weisstein (Feb. 11, 2005). This is roughly consistent with simple arguments based on the distribution of primes which suggest that only a single prime is expected in the first or so terms. The number of digits of the n term can be computed by noticing the pattern in the following table, where is the number of digits in n.

Therefore, the number of digits D(n) in the nth term can be written

where the second term is the repunit .

The results of concatenating the binary representations of the first few integers are 1, 110, 11011, 11011100, 11011100101, ... (Sloane's A058935). These digit sequences are plotted above for n = 1 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-dollar sequence).

The nth term of the reverse integer sequence consists of the concatenation of the first n positive integers written backwards: 1, 21, 321, 4321, ... (Sloane's A000422; Smarandache 1993, Dumitrescu and Seleacu 1994, Stephan 1998). The terms up to n = 9 are given by

The only prime in the first terms (Weisstein, Jun. 8, 2004) of this sequence is the 82nd term 828180...321 (Stephan 1998, Fleuren 1999), which has 155 digits. This is roughly consistent with simple arguments based on the distribution of primes which suggest that a single prime is expected in the first or so terms. The terms of the reverse integer sequence have the same number of digits as do the consecutive integer sequence.

There is an amazing connection between the numbers , , the Demlo numbers, and the repunits.

The concatenation of the first n primes gives 2, 23, 235, 2357, 235711, ... (Sloane's A019518; Smith 1996, Mudge 1997), sometimes called the Smarandache-Wellin numbers. This sequence converges to the digits of the Copeland-Erdos constant and is prime for terms 1, 2, 4, 128, 174, 342, 435, 1429, ... (Sloane's A046035; Ibstedt 1998, pp. 78-79; Crandall and Pomerance 2001, p. 72), with no others less than (Weisstein, Feb. 11, 2005). Therefore, the concatenation of all primes up to 2, 3, 7, 719, 1033, 2297, 3037, 11927, ... (Sloane's A046284) is prime.

The concatenation of the first n odd numbers gives 1, 13, 135, 1357, 13579, ... (Sloane's A019519; Smith 1996, Marimutha 1997, Mudge 1997). This sequence is prime for terms 2, 10, 16, 34, 49, 2570, ... (Sloane's A046036; Weisstein, Ibstedt 1998, pp. 75-76), with no others less than (Weisstein, Feb. 11, 2005). The corresponding primes are 13, 135791113151719, 135791113151719212325272931, ... (Sloane's A048847). The 2570th term, given by 1 3 5 7...5137 5139, has 9725 digits and was discovered by Weisstein in Aug. 1998.

The concatenation of the first n even numbers gives 2, 24, 246, 2468, 246810, ... (Sloane's A019520; Smith 1996; Marimutha 1997; Mudge 1997; Ibstedt 1998, pp. 77-78).

The concatenation of the first n square numbers gives 1, 14, 149, 14916, ... (Sloane's A019521; Marimutha 1997). The only prime in the first terms is the third term, 149, (Weisstein, Feb. 11, 2005).

The concatenation of the first n cubic numbers gives 1, 18, 1827, 182764, ... (Sloane's A019522; Marimutha 1997). There are no primes in the first terms (Weisstein, Feb. 11, 2005).

Champernowne Constant, Concatenation, Consecutive Numbers, Copeland-Erdos Constant, Cubic Number, Demlo Number, Even Number, Integer Sequence Primes, Odd Number, Smarandache Sequences, Square Number