SMARANDACHE PARTITION TYPE AND OTHER SEQUENCES*

Eng. Dr. Fanel IACOBESCU

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



Examples of Smarandache Partition type sequences:

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 = 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12
        = 12 + 12 + 12 + 12 + 12 +22
        = 12 + 22 + 22
        = 32,

     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 = 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13
        = 13 + 23,

     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(n1 ), f(n2 ), ..., f(nk ))

     for some k and n1 , n2 , ..., nk such that n1 + n2 + . . . + nk = 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 m1, m2, ..., mk be the first k terms of the sequence,
      where k >= 2; then mi , for i >= k+1, is the smallest number not equal
      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 m1 , m2, are the first two terms of the sequence,
      then mk , for k >= 3, is the smallest number such that no 3-term arithmetic
      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, ...

      Pn = 1 + p1 p2 . . . pn , where pk is the k-th prime.

      Question: How many of them are prime?

(7) Smarandache Square product sequence:

      2, 5, 37, 577, 14401, 518401, 25401601, 1625702401, 131681894401,
      13168189440001, 1593350922240001, ...

      Sn = 1 + s1 s2 . . . sn , where sk is the k-th square number.

      Question: How many of them are prime?

(8) Smarandache Cubic product sequence:

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

      Ck = 1 + c1 c2 ...cn , where ck is the k-th cubic number.

      Question: How many of them are prime?

(9) Smarandache Factorial product sequence:

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

      Fn = 1 + f1 f2 ...fn , where fk is the k-th factorial number.

      Question: How many of them are prime?

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

      Let un , n >= 1, be a positive integer sequence. Then we define a U-sequence as follows:

      Un = 1 + u1 u2 . . . un .

(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 m1 ,m2, are the first two terms of the sequence, then mk, for k >= 3,
      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, pn digits of "1", where pn is the n-th prime.

      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.