Smarandache sequences are any of a number of simply generated integer sequences resembling those considered in published works by Smarandache such as the consecutive number sequences and Euclid numbers (Iacobescu 1997). Some other "Smarandache" sequences are given below.
1. The decimal concatenation of 
copies of the positive integer 
, where 
: 1, 22, 333, 4444, 55555, ... (OEIS A000461;
Marimutha 1997). For 
, they have the simple formula
 |
(1)
|
where 
is a repunit. In general,
 |
(2)
|
where 
is the number of digits in 
. For 
, the 
th term is a multiple of 
greater than 
. The first term is 1, so no term is prime.
2. The concatenation of the first 
Fibonacci numbers: 1,
11, 112, 1123, 11235, ... (OEIS A019523; Marimutha
1997).
3. Starting with 1, 2, each subsequent term is the least integer strictly greater than the preceding term that is represented as a sum of the squares of two earlier terms at distinct positions: 1, 2, 5, 26, 29, 677, ... (OEIS A008318; Bencze 1997).
4. Starting with 1, 1, each subsequent term is the least integer strictly greater than the preceding term that is represented as a sum of the squares of one or more earlier terms at distinct positions: 1, 1, 2, 4, 5, 6, 16, 17, ... (OEIS A008319; Bencze 1997).
5. Starting with 1, 2, each subsequent term is the least integer strictly greater than the preceding term that cannot be represented as a sum of the squares of two earlier terms at distinct positions: 1, 2, 3, 4, 6, 7, 8, 9, 11, ... (OEIS A008320; Bencze 1997).
6. Starting with 1, each subsequent term is the least integer strictly greater than the preceding term that cannot be represented as a sum of the squares of one or more earlier terms at distinct positions: 1, 2, 3, 6, 7, 8, 11, ... (OEIS A008321; Bencze 1997).
7. Starting with 1, 2, each subsequent term is the least integer strictly greater than the preceding term that is represented as a sum of the cubes of two earlier terms at distinct positions: 1, 2, 9, 730, 737, ... (OEIS A008322; Bencze 1997).
8. Starting with 1, 1, each subsequent term is the least integer strictly greater than the preceding term that is represented as a sum of the cubes of one or more earlier terms at distinct positions: 1, 1, 2, 8, 9, 10, 512, 513, 514, ... (OEIS A019511; Bencze 1997).
9. Starting with 1, 2, each subsequent term is the least integer strictly greater than the preceding term that cannot be represented as a sum of the cubes of two earlier terms at distinct positions: 1, 2, 3, 4, 5, 6, 7, 8, 10, ... (OEIS A031980; Bencze 1997).
10. Starting with 1, each subsequent term is the least integer strictly greater than the preceding term that cannot be represented as a sum of the cubes of one or more earlier terms at distinct positions: 1, 2, 3, 4, 5, 6, 7, 10, 11, ... (OEIS A031981; Bencze 1997).
11. The number of partitions of 
, 1, 2, ... into positive square
numbers: 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 8, 9, 10, 10, 12, 13,
... (OEIS A001156; Iacobescu 1997).
12. The number of partitions of 
, 1, 2, ... into cubic numbers:
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, ... (OEIS A003108; Iacobescu 1997). The initial 1 in each
of these two partition sequences counts the empty
set of parts.
13. Two copies of the first 
positive integers: 11,
1212, 123123, 12341234, ... (OEIS A019524;
Iacobescu 1997).
14. Numbers written in base of triangular numbers, using the greedy algorithm with the positive values as place weights, choosing each coefficient as large as possible from the largest weight downward: 1, 2, 10, 11, 12, 100, 101, 102, 110, 1000, 1001, 1002, ... (OEIS A000462; Iacobescu 1997).
15. Numbers written in base of double factorial numbers, using the greedy algorithm with the positive values as place weights, choosing each coefficient as large as possible from the largest weight downward: 1, 10, 100, 101, 110, 200, 201, 1000, 1001, 1010, ... (OEIS A307102; Iacobescu 1997).
16. The increasing sequence starting with 1, 2 and constructed by a greedy algorithm, choosing each later term as the least integer strictly greater than the preceding term that creates no three-term arithmetic progression: 1, 2, 4, 5, 10, 11, 13, 14, 28, ... (OEIS A003278; Iacobescu 1997, Mudge 1997).
17. Numbers of the form 
for 
, 2, ...: 2, 5, 37, 577, 14401, 518401, 25401601, 1625702401,
131681894401, ... (OEIS A020549; Iacobescu
1997).
18. Numbers of the form 
for 
, 2, ...: 2, 9, 217, 13825, 1728001, 373248001, 128024064001,
... (OEIS A019514; Iacobescu 1997).
19. Numbers of the form 
for 
, 2, ...: 2, 3, 13, 289, 34561, 24883201, 125411328001, 5056584744960001,
... (OEIS A019515; Iacobescu 1997).
20. The increasing sequence starting with 1, 2 and constructed by a greedy algorithm, choosing each later term as the least integer strictly greater than the preceding term that creates no three-term geometric progression: 1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, ... (OEIS A000452; Iacobescu 1997).
21. Numbers repeating the digit 1 
times, where 
is the 
th prime: 11, 111, 11111, 1111111,
... (OEIS A031974; Iacobescu 1997). These are
a subset of the repunits.
22. Positive integers in increasing order with all 2s, 3s, 5s, and 7s (prime digits) removed: 1, 4, 6, 8, 9, 10, 11, 1, 1, 14, 1, 16, 1, 18, 19, 0, ... (OEIS A019516; Iacobescu 1997). Inputs leaving no digits after deletion are omitted in this and the following sequence.
23. Positive integers in increasing order with all 0s, 1s, 4s, and 9s (square digits) removed: 2, 3, 5, 6, 7, 8, 2, 3, 5, 6, 7, 8, 2, 2, 22, 23, ... (OEIS A031976; Iacobescu 1997).
24. Smarandache-Fibonacci triples; integers 
such that 
for 
, where 
is the Smarandache
function with 
: 3, 11, 121, 4902, 26245, ... (OEIS A015047;
Aschbacher and Mudge 1995; Ibstedt 1997, pp. 19-23; Begay 1997). The largest
known is 
.
25. Smarandache-Radu triplets; positive integers 
such that there are no primes
in the closed interval with endpoints 
and 
: 224, 2057, 265225, ... (OEIS A015048;
Radu 1994/1995, Begay 1997, Ibstedt 1997). The largest known is 
.
26. Smarandache crescendo sequence; integers obtained by concatenating strings of the first 
positive integers for
,
1, 2, ...: 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, ... (OEIS A002260;
Brown 1997, Brown and Castillo 1997). With indexing starting at 
, the 
th term is given by 
, where 
, with 
the floor function (Hamel
1997).
27. Smarandache descrescendo sequence; integers obtained by concatenating strings 
, 
, ..., 1 for 
, 2, ...: 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, ... (OEIS A004736;
Brown 1997, Smarandache 1997).
28. Smarandache crescendo pyramidal sequence, a.k.a. Smarandache descrescendo symmetric sequence; integers obtained by concatenating the blocks
1, 2, ..., 
,
,
..., 1 for 
,
2, ...: 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, ... (OEIS A004737;
Brown 1997, Brown and Castillo 1997, Smarandache 1997).
29. Smarandache descrescendo pyramidal sequence; integers obtained by concatenating the blocks 
, 
, ..., 1, 2, ..., 
for 
, 2, ...: 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4,
... (OEIS A004738; Brown 1997).
30. Smarandache crescendo symmetric sequence; concatenation of the blocks 1, 2, ..., 
,
,
,
..., 1 for 
,
2, ...: 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 2, 1, ... (OEIS A004739;
Brown 1997, Smarandache 1997).
31. Smarandache permutation sequence; numbers obtained by concatenating sequences of increasing length of increasing odd numbers and decreasing even numbers: 1, 2, 1, 3, 4, 2, 1, 3, 5, 6, 4, 2, ... (OEIS A004741; Brown 1997, Brown and Castillo 1997).
32. Smarandache pierced chain sequence; numbers of the form 
for 
,
1, ...: 101, 1010101, 10101010101, ... (OEIS A031982;
Ashbacher 1997). In addition, 
contains no primes
(Ashbacher 1997).
33. Smarandache symmetric sequence; decimal concatenation of the integers 1, 2, ..., ![[n/2]](/images/equations/SmarandacheSequences/Inline59.svg)
, 
, ..., 1 for 
, 2, ..., with an empty decreasing part for 
. Here ![[x]](/images/equations/SmarandacheSequences/Inline63.svg)
and 
are the ceiling function
and floor function: 1, 11, 121, 1221, 12321, 123321,
... (OEIS A007907; Smarandache 1993, Dumitrescu
and Seleacu 1994, sequence 3; Mudge 1995).
34. Smarandache square-digital sequence; positive square numbers all of whose digits are also squares: 1, 4, 9, 49, 100, 144, ... (OEIS A019544; Mudge 1997).
35. Square-digits; numbers composed of digits which are squares: 0, 1, 4, 9, 10, 11, 14, 19, 40, 41, ... (OEIS A046030).
36. Cube-digits; positive numbers composed of the digits 0, 1, and 8: 1, 8, 10, 11, 18, 80, 81, 88, 100, 101, ... (OEIS A046031).
37. Smarandache cube-digital sequence; cube-digit numbers which are themselves cubes: 1, 8, 1000, 8000, 1000000, ... (OEIS A019545; Mudge 1997).
38. Prime-digits; numbers composed of digits which are primes: 2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, ... (OEIS A046034).
39. Smarandache prime-digital sequence; prime-digit numbers which are themselves prime: 2, 3, 5, 7, 23, 37, 53, ... (OEIS A019546; Smith 1996, Mudge 1997). Dubner (2002) reported the following members of this sequence,
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a repunit and both 
and 
have 
digits.
40. Smarandache deconstructive sequence; integers constructed by sequentially repeating the digits 1-9 in the following way: 1, 23, 456, 7891, 23456, 789123, 4567891, ... (OEIS A007923; Smarandache 1993, Kashihara 1996, Ashbacher 1998, Atanassov 1999ab). Of these, 23, 4567891, 23456789, 1234567891, ... (OEIS A050234) are prime (Kashihara 1996, Ashbacher 1998).
