Consecutive number sequences are sequences constructed by concatenating numbers of a given type. Many of these sequences were considered by Smarandache and so are sometimes known as Smarandache sequences.
The most obvious consecutive number sequence is the sequence of the first 
positive integers joined
left-to-right, namely 1, 12, 123, 1234, ... (OEIS A007908;
Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998;
Wolfram 2002, p. 913).
In this work, members of this sequence will be termed Smarandache
numbers and the 
th such number written 
. No Smarandache primes 
exist for 
(Great Smarandache PRPrime search; Dec. 5, 2016).
The 
th
term 
of the "reverse integer sequence" consists of the concatenation of the
first 
positive integers written right-to-left: 1, 21,
321, 4321, ... (OEIS A000422; Smarandache 1993,
Dumitrescu and Seleacu 1994, Stephan 1998). The terms up to 
are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
The terms of the reverse integer sequence have the same number of digits as the consecutive integer sequence. The first few reverse integer concatenated primes occur for 
and 37765 (OEIS A176024), as summarized in
the following table. No other primes occur for 
(E. Weisstein, Nov. 10, 2015) or 
(Batalov, Nov. 9, 2015).
 | digits | discoverer |
| 82 | 155 | Stephan (1998), Fleuren (1999) |
| 37765 | 177719 | E. W. Weisstein (Apr. 6, 2010) |
There is an amazing connection between the numbers 
, 
, the Demlo numbers, and
the repunits.
The concatenation of the first 
primes is called a Smarandache-Wellin
number, the first few of which are 2, 23, 235, 2357, 235711, ... (OEIS A019518).
The first few Smarandache-Wellin primes
are 2, 23, 2357, ... (OEIS A069151), corresponding
to concatenations of the first 
, 2, 4, 128, 174, 342, 435, 1429 (OEIS A046035)
primes and having 1, 2, 4, 355, 499, 1171, 1543, 5719 (OEIS A263959)
decimal digits.
If digit sequences corresponding to concatenation of a right-truncated final prime are allowed, more primes are possible. In fact, these are precisely the Copeland-Erdős-constant primes, the first few of which are 2, 23, 2357, 23571113171, ... (OEIS A227529) and which have 1, 2, 4, 11, 353, 355, 499, 1171, 1543, 5719, 11048, 68433, 97855, 292447, ... decimal digits (OEIS A227530).
The concatenation of the first 
odd numbers gives 1, 13, 135,
1357, 13579, ... (OEIS A019519; Smith 1996,
Marimutha 1997, Mudge 1997). This sequence is prime
for terms 2, 10, 16, 34, 49, 2570, ... (OEIS A046036;
Weisstein, Ibstedt 1998, pp. 75-76), with no others less than 
(Weisstein, Oct. 9, 2015). The corresponding primes
are 13, 135791113151719, 135791113151719212325272931, ... (OEIS 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 
even numbers gives 2, 24, 246,
2468, 246810, ... (OEIS A019520; Smith 1996;
Marimutha 1997; Mudge 1997; Ibstedt 1998, pp. 77-78).
The concatenation of the first 
square numbers gives 1, 14,
149, 14916, ... (OEIS A019521; Marimutha 1997).
The only prime in the first 
terms is the third term, 149, (Weisstein, Oct. 9,
2015).
The concatenation of the first 
triangular numbers gives
1, 13, 136, 13610, ... (OEIS A078795). The
only primes in the first 
terms occur for terms 2 and 6, corresponding to primes
13 and 136101521 (OEIS A158750; Weisstein,
Oct. 9, 2015).
The concatenation of the first 
cubic numbers gives 1, 18,
1827, 182764, ... (OEIS A019522; Marimutha
1997). There are no primes in the first 
terms (Weisstein, Oct. 9, 2009).
The concatenation of the first 
Fibonacci numbers gives
1, 11, 112, 1123, 11235, ... (OEIS A019523;
Marimutha 1997). There are no primes in the first
1580 terms other than 
and 4, corresponding to primes 11 and 1123 (E. Weisstein,
Jul. 15, 2016).
