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