About Smarandache-Multiplicative Functions

Sabin Tabirca

Bucks University College, Computing Department, England


 The main objective in this note is to introduce the notion of S-multiplicative function and to give some simple properties concerning it. The name of S-multiplicative is a short of Smarandache-multiplicative and reflects the main equation of the Smarandache function.

Definition 1: A function f: N* -> N* is called S-multiplicative if:

(1). (a,b) = 1 => f(a * b) = max{ f(a), f(b) }

 The following functions are obviously S-multiplicative:
1. The constant function f :N* -> N*, f(n) = 1.
2. The Erdös function f :N* -> N*, f(n) = max{ p | p is prime and n :p}.[1]
3. The Smarandache function S:N* -> N, S(n) = max{ p| p! : n}.[3]

Certainly, many properties of multiplicative functions[2] can be translated for S-multiplicative functions. The main important property of this function is presented in the following.

Definition 2: If  f :N* -> N is a function, then

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f : N* -> N  is defined by
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f(n) = min{ f(d)  | n:d }.


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Theorem 1: If f is S-multiplicative function, then f is S-multiplicative.


Proof: This proof is made using the following simple remark:

(2). (d|(a * b) /\ (a,b) = 1 ) => ((Ed1 | a)(Ed2 | b)(d1, d2) = 1 /\ d = d1 * d2)

If d1 and d2 satisfy (2), then f(d1 * d2) = max { f(d1) , f(d2) }. Let a,b be two natural numbers, such that (a,b) = 1. Therefore, we have


_
(3) f(a * b) =   min  f(d)  =     min    f(d1 , d2)  =   min   min   max  { f( d1) ,f( d2)}.
 d|a*b d1|a,d2|a d1|a  d2|a

Applying the distributing property of the max and min functions, equation (3) is transformed as follows:

_ _ _
f(a * b) = max {  min  f(d1) ,  min  f(d2) }  =  max { f(a) , f(b) }. Therefore,
 d1|a  d2|a
_
the function f is S- multiplicative.


 We believe that many other properties can be deduced for S-multiplicative functions. Therefore, it will be in our attention to further investigate these functions.

References

[1] Erdös, P.:(1974) Problems and Result in Combinatorial Number Theory, Bordaux.
[2] Hardy, G. H. and Wright, E. M.:(1979) An Introduction to Number Theory, Clarendon Press, Oxford.
[3] F. Smarandache: (1980) 'A Function in Number Theory', Analele Univ. Timisoara, XVIII.