**Electrotechnic Faculty of Craiova, Romania**

**ABSTRACT**

Thanks to C. Dumitrescu and Dr. V. Seleacu of the |

University of Craiova, Department of Mathematics, |

I became familiar with some of the Smarandache |

Sequences. I list some of them, as well as questions |

related to them. Now I'm working in a few conjectures |

A. 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, ....

(How many times is n written as a sum of non-null squares, disregarding the order of the terms:

for example:

9 = 1

therefore ns(9) = 4.)

B. 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, ...

(How many times is n written as a sum of non-null cubes, disregarding the order of the terms: for example:

9 = 1

therefore, nc(9) = 2.)

C. General-partition type sequence:

Let f be an arithmetic function and R a relation among numbers. (How many times can n be written

under the form:

n = R(f(n

for some k and n

Examples of other sequences:

(1) Smarandache Anti-symmetric sequence:

11, 1212, 123123, 12341234, 1234512345, 123456123456,

12345671234567, 1234567812345678, 123456789, 123456789,

1234567891012345678910, 1234567891011, 1234567891011, ...

(2) Smarandache Triangular base:

1, 2, 10, 11, 12, 100, 101, 102, 110, 1000, 1001, 1002, 1010, 1011,

10000, 10001, 10002, 10010, 10011, 10012, 100000, 100001, 100002,

100010, 100011, 100012, 100100, 1000000, 1000001, 1000002, 1000010,

1000011, 1000012, 1000100, ...

(Numbers written in the triangular base, defined as follows:

t(n) = (n(n+1))/2, for n >= 1.)

(3) Smarandache Double factorial base:

1, 10, 100, 101, 110, 200, 201, 1000, 1001, 1010, 1100, 1101, 1110,

1200, 10000, 10001, 10010, 10100, 10101, 10110, 10200, 10201, 11000,

11001, 11010, 11100, 11101, 11110, 11200, 11201, 12000, ...

(Numbers written in the double factorial base, defined as follows:

df(n) = n!!)

(4) Smarandache Non-multiplicative sequence:

General definition: Let m

where k >= 2; then m

to the product of k previous distinct terms.

(5) Smarandache Non-arithmetic progression:

1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40, 41, 64, ...

General definition: if m

then m

progression is in the sequence. In our case the first two terms are 1, respectively 2.

Generalization: same initial conditions, but no i-term arithmetic progression

in the sequence (for a given i >= 3).

(6) Smarandache Prime product sequence:

2, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131,

7420738134811, 304250263527211, ...

P

Question: How many of them are prime?

(7) Smarandache Square product sequence:

2, 5, 37, 577, 14401, 518401, 25401601, 1625702401, 131681894401,

13168189440001, 1593350922240001, ...

S

Question: How many of them are prime?

(8) Smarandache Cubic product sequence:

2, 9, 217, 13825, 1728001, 373248001, 128024064001, 65548320768001, ...

C

Question: How many of them are prime?

(9) Smarandache Factorial product sequence:

2, 3, 13, 289, 34561, 24883201, 125411328001, 5056584744960001, ...

F

Question: How many of them are prime?

(10) Smarandache U-product sequence {generalization}:

Let u

U

(11) Smarandache Non-geometric progression.

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24,

26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 47,

48, 50, 51, 53, . . .

General definition: if m

is the smallest number such that no 3-term geometric progression is in the sequence.

In our case the first two terms are 1, respectively 2.

(12) Smarandache Unary sequence:

11, 111, 11111, 1111111, 11111111111, 1111111111111, 1111111111111111,

1111111111111111111, 11111111111111111111111,

11111111111111111111111111111, 1111111111111111111111111111111, ...

u(n) = 11...1, p

The old question: are there are infinite number of primes belonging to the sequence?

(13) Smarandache No-prime-digit sequence:

1, 4, 6, 8, 9, 10. 11, 1, 1, 14, 1, 16. 1, 18, 19, 0, 1, 4, 6, 8, 9,

0, 1, 4, 6, 8, 9, 40, 41, 42, 4, 44, 4, 46, 48, 49, 0, ...

(Take out all prime digits of n.)

(14) Smarandache No-square-digit-sequence.

2, 3, 5, 6, 7, 8, 2, 3, 5, 6, 7, 8, 2, 2, 22, 23, 2, 25, 26, 27, 28,

2, 3, 3, 32, 33, 3, 35, 36, 37, 38, 3, 2, 3, 5, 6, 7, 8, 5, 5, 52, 53,

5, 55, 56, 57, 58, 5, 6, 6, 62, ...

(Take out all square digits of n.)

* This paper first appeared in Bulletin of Pure and Applied Sciences, Vol. 16 E(No. 2) 1997; P. 237-240.