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Smarandache Function Journal, Vol. 6, No. 1-2-3, 1995, pp. 55-58. 


SMARANDACHE FUNCTIONS OF THE SECOND KIND 
by Ion Bălăcenoiu and Constantin Dumitrescu 


Departament of Mathematics, University of Craiova 
Craiova (1100), Romania 


The Smarandache functions of the second kind are defined in [1] thus: 
SN oN, S*(n)= S (k) forneN, 


where S, are the Smarandache functions of the first kind (see [3]). 

We remark that the function S' has been defined in [4] by F. Smarandache because 
Si=S. 

Let, for example, the following table with the values of S°: 


n |1 2 3 4 5 6 7 8 9 10 11 12 #13 14 
vm 11 4 6 6 10 6 14 12 12 10 22 8 2 14 


Obviously, these functions S* aren't monotony, aren't periodical and they have fixed 
points. 


1. Theorem. For k,n EN is true S*(n)<n-k. 
Proof. Let n= py'pit...pe and S(n) = max{S, (a,)} = S(p% 


Because S*(n) = S(n*) = max{S, (ak) } = S(po*) < KS (pe) < kS (py) = KS(n) 
and S(n)<n, [see [3]], it results: 


(1) S*(n)<n-k for every n,k EN’. 
2. Theorem. All prime numbers p 2 5 are maximal points for S* , and 
S*(p) = p[k -i ,(k)], where ospa |E 


Proof. Let p25 be a prime number. Because S,-(k) <S, (k), Spa (k) <S, (k) [see 


[2] it results that S*(p-1)< S*(p) and S*(p+1)<S*(p), so that S*(p) isa relative 
maximum value. 


55 


Obviously, 


(2) S*(p)=S,(k)= pik -ip(k)) with osoo E] 
3)  S*(p)=pk for p2k. 
3. Theorem. The mumbers kp. for p prime and p>k are the fixed points of S*. 


Proof. Let p be a prime number, m= p;"...p;' be the prime factorization of m and 
p>max{m,k}. Then pa, <p <p for iecelt, therefore we have: 


S*(m- p) = SU(mp)¥ = max{S,o,.5p(#)} = S0) = Wp. 


For m=k we obtain: 


S* (kp) = kp so that kp isa fixed point. 


4. Theorem. The functions S* have the following properties: 


S* =0 (n'**), for e>0 




















k 
lim sup = Sa 
nro n 
Proof. Obviously, 
k 
0< tim E = tim S02 < tim S@ = btm “= 0 for 
ae nts ae ni** BEROEN nt" aso re 
S=0 (n'**), — [see[4]]. 
Therefore we have S* =0 (n'**), and: 
k 
T (") im sups - im 267 ) -k 
n=% n ne n | gach Pp 


56 


5. Theorem, [see[1]]. The Smarandache functions of the second kind standardise 
(N’,-) in (N’,s,+) by: 


55, max{S* (a), S(b)} < S* (ab) < S*(a) + S* (b) 
and (N,-) in (N’,<,-) by: 


2a: max{5* (a), S* (b)} < S* (ab) < S“ (a): S* (b) for every a,b eN’ 


6. Theorem. The functions S* are, generally speaking, increasing. It means that: 
Yn EN =m EN’ so that Ym> m => S*(m)2> S*(n) 
Proof. The Smarandache function is generally increasing, [see [4]], it means that : 
(3) VteN an(t)eN’ sothat Vr2>n => S(r)2>S(t) 


Let t=n" and r=7,(t) so that Vr>n => S(r)2S(n*). 

Let m =| {ry [+ 1. Obviously m > Yn > m zand m>m > m > né. 
Because m‘ >m) 2r it results S(m*)>S(n*) or S*(m)2S*(n). 
Therefore 


VneN am = [Um | +! so that 


vm2zm => S*(m)2S*(n) where 1 =1,(n*) 


is given from (3). 


7. Theorem. The function S* has its relative minimum values for every n= p!, where p 
is a prime number and p > max{3,k}. 


Proof. Let p!= p!-p}--p”-p be the canonical decomposition of p!, where 
2 = Pi <3= pP << Pma < p. Because p! is divisible by p/ it results S(p?) < p=S(p) for 


every jJ €l,m. 
Obviously, 


S*(p!) = Sl(p)*}= max{s(p**),S(p*)] 
Because S{ p*”} < kS(p!) < kS(p) = kp = S(p*) for k < p, it results that we have 


(4) S“(p!) = S(p*)=kp, for k <p 
57 


Let p!-l=q)-q?---q) be the canonical decomposition for pl!-1, then 
q,; > p forj elt. 
It follows S( p!— 1) = max{S(q/)} = S(q's) with q„ > P. 
J 


Because S(q") > S(p) = S(p!) it results S(p!- 1) > S(p!). 
Analogous it results S( p!+1)> S(p!). 
Obviously 


(5) Spl- = Ship- 1)" ] > Stk") 2 SIE) > S(p) = kp 


(6) S*(p!+1)= Sl(pi+)*] >k-p 


For p2max{3,k} out of (4), (5), (6) it results that p! are the relative minimum 
points of the functions S*. 


REFERENCES 


[1] L Balicenoiu, Smarandache Numerical functions, Smarandache Function Journal, 
vol. 4-5, no.1, (1994), p.6-13. 


[2] L B&licenoiu, The monotony of Smarandache functions of first kind., Smarandache 
Function Journal, vol.6, 1995. 


[3] L Balicenoiu, V. Seleacu, Some properties of Smarandache functions of the type I, 
Smarandache Function Journal, vol.6, 1995. 


[4] F. Smarandache, A function in the Number Theory. An.Univ.Timisoara, seria 
st.mat. Vol. XVII, fasc. 1, p.79-88, 1980.