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

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 

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 

1. Theorem. For k,n EN is true S*(n)<n-k. 
Proof. Let n= py' 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. 



(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 

lim sup = Sa 
nro n 
Proof. Obviously, 
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: 
T (") im sups - im 267 ) -k 
n=% n ne n | gach Pp 


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). 

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. 

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 

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. 

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

(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*. 


[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.