FUNCTIONS IN NUMBER THEORY


1) Smarandache-Kurepa Function:
     For p prime, SK(p) is the smallest integer such that !SK(p) is
     divisible by p, where !SK(p) = 0! + 1! + 2! + ... + (p-1)!

    For example:
      p     2  3  7  11  17  19  23  31  37  41  61  71  73  89
     SK(p)  2  4  6   6   5   7   7  12  22  16  55  54  42  24

Reference:
 [1] C.Ashbacher, "Some Properties of the Smarandache-Kurepa and
     Smarandache-Wagstaff Functions", in <Mathematics and Informatics
     Quarterly>, Vol. 7, No. 3, pp. 114-116, September 1997.



2) Smarandache-Wagstaff Function:
     For p prime, SW(p) is the smallest integer such that W(SW(p)) is
     divisible by p, where W(p) = 1! + 2! + ... + (p)!

    For example:
      p     3  11  17  23  29  37  41  43  53  67  73  79  97
     SW(p)  2   4   5  12  19  24  32  19  20  20   7  57   6

Reference:
 [1] C.Ashbacher, "Some Properties of the Smarandache-Kurepa and
     Smarandache-Wagstaff Functions", in <Mathematics and Informatics
     Quarterly>, Vol. 7, No. 3, pp. 114-116, September 1997.



3)  Smarandache Ceil Functions of k-th Order:
      Sk(n) is the smallest integer for which n divides Sk(n)^k.

     For example, for k=2, we have:
      n     1  2  3  4   5  6  7  8   9  10 11 12  13  14  15  16
     S2(n)  2  4  3  6  10 12  5  9  14  8  6  20  22  15  12   7

References:
 [1] H.Ibstedt, "Surfing on the Ocean of Numbers -- A Few Smarandache
     Notions and Similar Topics", Erhus Univ. Press, Vail, USA, 1997;
     pp. 27-30.
 [2] A.Begay, "Smarandache Ceil Functions", in <Bulletin of Pure and
     Applied Sciences>, India, Vol. 16E, No. 2, 1997, pp. 227-229.



4)  Pseudo-Smarandache Function:
     Z(n) is the smallest integer such that 1 + 2 + ... + Z(n) is divisible
     by n.

     For example:
      n    1   2   3   4   5    6    7
     Z(n)  1   3   2   3   4    3    6
  
Reference:
 [1] K.Kashihara, "Comments and Topics on Smarandache Notions and
     Problems", Erhus University Press, Vail, USA, 1996.



5) Smarandache Near-To-Primordial Function:
                                                  *       *       *
   SNTP(n) is the smallest prime such that either p  - 1, p  , or p  + 1
   is divisible by n,
          *
   where p , of a prime number p, is the product of all primes less than
   or equal to p.

   For example:

   n          1  2  3  4  5  6  7  8  9  10  11  ...  59 ...

   SNTP(n)    2  2  2  5  3  3  3  5  ?   5  11  ...  13  ...

References:
 [1] Mudge, Mike, "The Smarandache Near-To-Primordial (S.N.T.P.) Function",
     <Smarandache Notions Journal>, Vol. 7, No. 1-2-3, August 1996, p. 45.
 [2] Ashbacher, Charles, "A Note on the Smarandache Near-To-Primordial
     Function", <Smarandache Notions Journal>, Vol. 7, No. 1-2-3, August 1996, 
     pp. 46-49.



 6)  Smarandache Double-Factorial Function:
     SDF(n) is the smallest number such that SDF(n)!! is divisible by n,
     where the double factorial 
         m!! = 1x3x5x...xm, if m is odd;
     and m!! = 2x4x6x...xm, if m is even.
     
     For example:
     
     n         1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16

     SDF(n)    1  2  3  4  5  6  7  4  9  10  11   6  13  14   5   6

Reference:
 [1[] Dumitrescu, C., Seleacu, V., "Some notions and questions in number
      theory", Erhus Univ. Press, Glendale, 1994, Section #54 ("Smarandache
      Double Factorial Numbers").



  7)  Smarandache Primitive Functions:

      Let p be a positive prime.
                                                                         n
      S  : N ---> N,  having the property that (S (n))! is divisible by p  ,
       p                                         p

      and it is the smallest integer with this property.

      For example:

      S (4) = 9, because 9! is divisible by 3^4, and it is the smallest one 
       3
      with this property.

      These functions help computing the Smarandache Function.

Reference:
 [1] Smarandache, Florentin, "A function in number theory", <Analele
     Universitatii Timisoara>, Seria St. Mat., Vol. XVIII, fasc. 1, 1980, 
     pp. 79-88.


  
  8)  Smarandache Function:
      
      S : N ---> N, S(n) is the smallest integer such that S(n)! is
      divisible by n.

Reference:
 [1] Smarandache, Florentin, "A function in number theory", <Analele
     Universitatii Timisoara>, Seria St. Mat., Vol. XVIII, fasc. 1, 1980, 
     pp. 79-88.


  9) Smarandache Functions of the First Kind:

           *      *
     S  : N  --> N 
      n
                r
    i)  If n = u   (with u = 1, or u = p prime number), then

         S (a) = k, where k is the smallest positive integer such that
          n
                              ra
         k! is a multiple of u  ;
                 
                 r1    r2       rt
    ii) If n = p1  . p2   ... pt  , then S (a) =   max   { S      (a) }.
                                          n      1<=j<=t       rj
                                                             pj


  10) Smarandache Functions of the Second Kind:

       k    *      *     k                      *
      S  : N  --> N ,   S (n) = S (k) for k in N ,
                                 n

      where S  are the Smarandache functions of the first kind.
             n



  11) Smarandache Function of the Third Kind:

       b
      S (n) = S  (b ),  where S   is the Smarandache function of the
       a       a   n           a
                n               n

      first kind, and the sequences (a ) and (b ) are different from
                                      n        n

      the following situations:
                                         *
      i)  a  = 1  and  b  = n, for n in N ;
           n            n
                                         *
      ii) a  = n  and  b  = 1, for n in N .
           n            n


Reference:
 [1] Balacenoiu, Ion, "Smarandache Numerical Functions", <Bulletin of Pure
     and Applied Sciences>, Vol. 14E, No. 2, 1995, pp. 95-100.