Consecutive Number Sequences

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).

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