Consecutive number sequences are sequences constructed by the concatenation of 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 
, for a positive integer 
.
In the search for Smarandache primes, WraithX
(2025) reported no probable primes 
for 
on Mar. 23, 2025.
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 for 
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 reported reverse integer concatenated primes
or probable primes occur for 
and 37765 (OEIS A176024),
as summarized in the following table. The 37765th term is reported as a probable
prime. No additional such terms are reported through 
(OEIS A176024).
 | digits | discoverer |
| 82 | 155 | Stephan (1998), Fleuren (1999) |
| 37765 | 177719 | E. W. Weisstein (Apr. 6, 2010) |
For 
,
the Demlo numbers satisfy 
, where 
and 
is the 
th repunit.
The increasing and decreasing constructions extend to any integer base 
. Let 
be the integer obtained by
concatenating the base-
representations of 1, 2, ..., 
, and let 
use the blocks 
, 
, ..., 1, for 
. In the decreasing construction, each block retains
its usual digit order. For example, 
and 
. Similarly, 
and 
.
The following table distinguishes the concatenated digit strings, recorded as decimal integers, from their values when evaluated in the original
base. Except for an additional zeroth term 0 in OEIS A058935, the listed constructions start at 
.
| base | block order | digit strings | integer values |
| 2 | increasing | A058935 | A047778 |
| 2 | decreasing | A300570 | A098780 |
| 3 | increasing | A360502 | A048435 |
| 3 | decreasing | A360505 | A360506 |
Reported prime or probable prime terms of 
occur at indices 2, 5, 82, 2546, ... (OEIS A360503),
and those of 
at 2, 5, 13, 57, 109, 638, 3069, ... (OEIS A360507).
Reading the decreasing binary strings directly as decimal integers instead gives prime
or probable prime terms at indices 2, 4, 13, 57,
64, 349, ... (OEIS A360508). These are distinct
primality testing problems. For example, 
is prime, while the
same digit string read as the
decimal integer 11101 is
composite.
In base 10, concatenating the increasing blocks 1, 2, ..., 
and then the decreasing blocks 
, ..., 1 gives the odd-index terms of the Smarandache
symmetric sequence. This is the other construction featured in Numberphile (2021),
alongside the search for Smarandache primes.
The related binary, ternary,
and other concatenation searches are collected in the OEIS index (OEIS Foundation).
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 reported Smarandache-Wellin
primes or probable 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. Rivera reports the 5719-digit term as a probable prime (Rivera).
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). The reported lengths for primes and probable primes are 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). E. W. Weisstein (Jul. 28, 2016) reported no primes
in the first 1891 terms other than 
and 4, corresponding to primes
11 and 1123.
The concatenation of the first 
terms of the prime counting
function gives 0, 01, 012, 0122, 01223, 012233, 0122334, ... (Campbell 2024).
Interpreting the limiting string as decimal
digits gives the prime-counting
concatenation constant, whose successive digits are
enumerated by OEIS A366033.
