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). Other Smarandache-type sequences are given below.

- 1. The concatenation of copies of the Integer : 1, 22, 333, 4444, 55555, ... (Sloane's A000461; Marimutha 1997),
- 2. The concatenation of the first Fibonacci Numbers: 1, 11, 112, 1123, 11235, ... (Sloane's A019523; Marimutha 1997),
- 3. The smallest number that is the sum of squares of
*two*distinct earlier terms: 1, 2, 5, 26, 29, 677, ... (Sloane's A008318, Bencze 1997), - 4. The smallest number that is the sum of squares of any number of distinct earlier terms: 1, 1, 2, 4, 5, 6, 16, 17, ... (Sloane's A008319, Bencze 1997),
- 5. The smallest number that is
*not*the sum of squares of*two*distinct earlier terms: 1, 2, 3, 4, 6, 7, 8, 9, 11, ... (Sloane's A008320, Bencze 1997), - 6. The smallest number that is
*not*the sum of squares of any number of distinct earlier terms: 1, 2, 3, 6, 7, 8, 11, ... (Sloane's A008321, Bencze 1997), - 7. The smallest number that is a sum of cubes of
*two*distinct earlier terms: 1, 2, 9, 730, 737, ... (Sloane's A008322, Bencze 1997), - 8. The smallest number that is a sum of cubes of any number of distinct earlier terms: 1, 1, 2, 8, 9, 10, 512, 513, 514, ... (Sloane's A019511, Bencze 1997),
- 9. The smallest number that is
*not*a sum of cubes of*two*distinct earlier terms: 1, 2, 3, 4, 5, 6, 7, 8, 10, ... (Sloane's A031980, Bencze 1997), - 10. The smallest number that is
*not*a sum of cubes of any number of distinct earlier terms: 1, 2, 3, 4, 5, 6, 7, 10, 11, ... (Sloane's A031981, Bencze 1997), - 11. The number of Partitions of a number , 2, ... into Square Numbers: 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 8, 9, 10, 10, 12, 13, ... (Sloane's A001156, Iacobescu 1997),
- 12. The number of Partitions of a number , 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, ... (Sloane's A003108, Iacobescu 1997),
- 13. Two copies of the first Positive integers: 11, 1212, 123123, 12341234, ... (Sloane's A019524, Iacobescu 1997),
- 14. Numbers written in base of triangular numbers: 1, 2, 10, 11, 12, 100, 101, 102, 110, 1000, 1001, 1002, ... (Sloane's A000462, Iacobescu 1997),
- 15. Numbers written in base of double factorial numbers: 1, 10, 100, 101, 110, 200, 201, 1000, 1001, 1010, ... (Sloane's A019513, Iacobescu 1997),
- 16. Sequences starting with terms which contain no three-term arithmetic progressions starting with : 1, 2, 4, 5, 10, 11, 13, 14, 28, ... (Sloane's A003278, Iacobescu 1997, Mudge 1997, Weisstein),
- 17. Numbers of the form : 2, 5, 37, 577, 14401, 518401, 25401601, 1625702401, 131681894401, ... (Sloane's A020549, Iacobescu 1997),
- 18. Numbers of the form : 2, 9, 217, 13825, 1728001, 373248001, 128024064001, ... (Sloane's A019514, Iacobescu 1997),
- 19. Numbers of the form : 2, 3, 13, 289, 34561, 24883201, 125411328001, 5056584744960001, ... (Sloane's A019515, Iacobescu 1997),
- 20. Sequences starting with terms which contain no three-term geometric progressions starting with : 1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, ... (Sloane's A000452, Iacobescu 1997),
- 21. Numbers repeating the digit 1 times, where is the th prime: 11, 111, 11111, 1111111, ... (Sloane's A031974, Iacobescu 1997). These are a subset of the Repunits,
- 22. Integers 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, ... (Sloane's A019516, Iacobescu 1997),
- 23. Integers 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, ... (Sloane's A031976, Iacobescu 1997).
- 24. (Smarandache-Fibonacci triples) Integers such that , where is the Smarandache Function: 3, 11, 121, 4902, 26245, ... (Sloane's A015047; Aschbacher and Mudge 1995; Ibstedt 1997, pp. 19-23; Begay 1997). The largest known is 19,448,047,080,036,
- 25. (Smarandache-Radu triplets) Integers such that there are no primes between the smaller and larger of and : 224, 2057, 265225, ... (Sloane's A015048; Radu 1994/1995, Begay 1997, Ibstedt 1997). The largest known is 270,329,975,921,205,253,634,707,051,822,848,570,391,313,
- 26. (Smarandache crescendo sequence): Integers obtained by concatenating strings of the first integers for , 1, 2, ...: 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, ... (Sloane's A002260; Brown 1997, Brown and Castillo 1997). The th term is given by , where , with the Floor Function (Hamel 1997),
- 27. (Smarandache descrescendo sequence): Integers obtained by concatenating strings of the first integers for , 2, 1: 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, ... (Sloane's A004736; Smarandache 1997, Brown 1997),
- 28. (Smarandache crescendo pyramidal sequence, a.k.a. Smarandache descrescendo symmetric sequence): Integers obtained by concatenating strings of rising and falling integers: 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, ... (Sloane's A004737; Brown 1997, Brown and Castillo 1997, Smarandache 1997),
- 29. (Smarandache descrescendo pyramidal sequence): Integers obtained by concatenating strings of falling and rising integers: 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, ... (Sloane's A004738; Brown 1997),
- 30. (Smarandache crescendo symmetric sequence): 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 2, 1, ... (Sloane's 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, ... (Sloane's A004741; Brown 1997, Brown and Castillo 1997),
- 32. (Smarandache pierced chain sequence): Numbers of the form for , 1, ...: 101, 1010101, 10101010101, ... (Sloane's A031982; Ashbacher 1997). In addition, contains no Primes (Ashbacher 1997),
- 33. (Smarandache symmetric sequence): 1, 11, 121, 1221, 12321, 123321, ... (Sloane's A007907; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 3; Mudge 1995),
- 34. (Smarandache square-digital sequence): square numbers all of whose digits are also squares: 1, 4, 9, 49, 100, 144, ... (Sloane's A019544; Mudge 1997),
- 35. (Square-digits): numbers composed of digits which are squares: 0, 1, 4, 9, 10, 11, 14, 19, 40, 41, ... (Sloane's A046030),
- 36. (Cube-digits): numbers composed of digits which are cubes: 1, 8, 10, 11, 18, 80, 81, 88, 100, 101, ... (Sloane's A046031),
- 37. (Smarandache cube-digital sequence): cube-digit numbers which are themselves cubes: 1, 8, 1000, 8000, 1000000, ... (Sloane's A019545; Mudge 1997),
- 38. (Prime-digits): numbers composed of digits which are primes: 2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, ... (Sloane's A046034),
- 39. (Smarandache prime-digital sequence): prime-digit numbers which are themselves prime: 2, 3, 5, 7, 23, 37, 53, ... (Sloane's A019546; Smith 1996, Mudge 1997).

**References**

Aschbacher, C. *Collection of Problems On Smarandache Notions.* Vail, AZ: Erhus University Press, 1996.

Aschbacher, C. and Mudge, M. *Personal Computer World.* pp. 302, Oct. 1995.

Begay, A. ``Smarandache Ceil Functions.'' *Bull. Pure Appl. Sci.* **16E**, 227-229, 1997.

Bencze, M. ``Smarandache Recurrence Type Sequences.'' *Bull. Pure Appl. Sci.* **16E**, 231-236, 1997.

Bencze, M. and Tutescu, L. (Eds.). *Some Notions and Questions in Number Theory, Vol. 2.*
http://www.gallup.unm.edu/~smarandache/SNAQINT2.TXT.

Brown, J. ``Crescendo & Descrescendo.'' In *Richard Henry Wilde: An Anthology in Memoriam (1789-1847)* (Ed. M. Myers).
Bristol, IN: Bristol Banner Books, p. 19, 1997.

Brown, J. and Castillo, J. ``Problem 4619.'' *School Sci. Math.* **97**, 221-222, 1997.

Dumitrescu, C. and Seleacu, V. (Ed.). *Some Notions and Questions in Number Theory, 4th ed.*
Glendale, AZ: Erhus University Press, 1994.
http://www.gallup.unm.edu/~smarandache/SNAQINT.TXT.

Dumitrescu, C. and Seleacu, V. (Ed.).
*Proceedings of the First International Conference on Smarandache Type Notions in Number Theory.*
Lupton, AZ: American Research Press, 1997.

Hamel, E. Solution to Problem 4619. *School Sci. Math.* **97**, 221-222, 1997.

Iacobescu, F. ``Smarandache Partition Type and Other Sequences.'' *Bull. Pure Appl. Sci.* **16E**, 237-240, 1997.

Ibstedt, H. *Surfing on the Ocean of Numbers--A Few Smarandache Notions and Similar Topics.*
Lupton, AZ: Erhus University Press, 1997.

Kashihara, K. *Comments and Topics on Smarandache Notions and Problems.*ail, AZ: Erhus University Press, 1996.

Mudge, M. ``Top of the Class.'' *Personal Computer World,* 674-675, June 1995.

Mudge, M. ``Not Numerology but Numeralogy!'' *Personal Computer World,* 279-280, 1997.

*Programs and the Abstracts of the First International Conference on Smarandache Notions in Number Theory.* Craiova, Romania, Aug. 21-23, 1997.

Radu, I. M. *Mathematical Spectrum* **27**, 43, 1994/1995.

Sloane, N. J. A. Sequences A000452, A000461, A000462, A001156/M0221, A002260, A003108/M0209, A003278/M0975, A004736, A004737, A004738, A004739, A004741, A007907, A008318, A008319, A008320, A0083201, A008322, A015047, A015048, A019524, A019511, A019513, A019514, A019515, A019516, A019523, A019544, A019545, A019546 A020549, A031974, A031976, A031980, A031981, A031982, A046030, A046031, and A046034 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Smarandache, F. ``Properties of the Numbers.'' Tempe, AZ: Arizona State University Special Collection, 1975.

Smarandache, F. *Only Problems, Not Solutions!, 4th ed.* Phoenix, AZ: Xiquan, 1993.

Smarandache, F. *Collected Papers, Vol. 2.* Kishinev, Moldova: Kishinev University Press, 1997.

Smith, S. ``A Set of Conjectures on Smarandache Sequences.'' *Bull. Pure Appl. Sci.* **15E**, 101-107, 1996.

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