Smarandache functions

[CrystyB]

code:

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 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





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





3)  Smarandache Ceil Functions of n-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





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





5) Smarandache Near-To-Primordial Function:


                                                  *       *       *


   SNTP(n) is the smalest 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  ...





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





 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.





 8)  Smarandache Function:


      


      S : N ---> N, S(n) is the smallest integer such that S(n)! is


      divisible by n.





 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

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Hope this helps as a reference (it took me a long time to find this text through all those links...)

[CrystyB]

mith, i wonder how do you pronounce the last five letter of his name: dash, or da-ke (as in DAkota, KEntucky)?

Second, i have a few stuff to ask if you don't mind:
SK(3) is said to be 4, [but !4=10] and i think there is no such number. I think i can prove that too.
i don't get the meaning of Sk(n). The examples didn't help me either.
there is no SNTP(4x) for any integer x. I think i can prove that too.
is it easy to show that Z(F(p))=p?
where did you found the problem of Z(n)=Z(n+1) instead of S(...)?

TYVMia,
Cristian

[CrystyB]

mith, have you read this topic anytime during the last year?

(IOW bump)