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Smarandache Notions Journal, Vol. 7, No. 1-2-3, 1996, pp. 39-45. 


THE MONOTONY OF SMARANDACHE FUNCTIONS 
OF FIRST KIND 


by Ion Bălăcenoiu 


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


Smarandache functions of first kind are defined in [1] thus: 
SS:N >N’, S,(k)=1 and Sn (k) = max {S, (i,4)}, 
Isjsr "J 
where n= p} - p? --- p! and Sp, are functions defined in [4]. 
They È,- standardise (N°,+) in (N",<,+) in the sense that 
Z: max{S,(a),5,(5)} < S,(a+b) < S,(a)+S,(b) 
for everya,beN” and Z,- standardise (N,+) in (N’,<,-) by 
Z: max{S,(a),5,(b)} < S,(a+b) < S,(a)-S,(b), for every a,b e N° 


In [2] it is prooved that the functions S, are increasing and the sequence {S f heyt B 
aiso increasing. It is also proved that if p,q are prime numbers, then 


pi<q>S, <S, and i <q >S <S 


where i e N°. 
It would be used in this paper the formula 


S,(k)= p(k -i ), for same i, satisfying osas E], (see [3]) (1) 


1. Proposition. Let p be a prime number and k,,k, e N°. Ifk <k, then i, <i,,, 
where i, ‚i are defined by (1). 


Proof. It is known that $,:N° —>N° and S,(k)= pk for k< p. If S, (k) = mp" 
with m, œ e N° ,(m, p) = 1, there exist a consecutive numbers: 
n,n+1,...n+a-l so that 
k e{n,n+1,...,n+a-1} and 
S, (n) =S,(n+1) =-= S(n+a-!1), 


39 


this means that S, is stationed the a—1 steps (k — k +1). 

If k <k, and S,(k)=S,(k,), because S,(k)= p(k, -ik), S,(k:) = p(k, ik) 
it results i, <i, . 

If k <k, and S,(k,)<S,(k,), it is easy to see that we can write: 


iy, =f, +2Z(a-1) 


mp” < S (k) 
then 2, € {0,1,2,...,æ- 1} 
and 


where A =0 for S (k) = mp", if S,(%) = mp* 


i, = By +> (a-1) 


mp“ <5, (ta) 
A, €{0,1,2,...,a-1}. 
Now is obviously that k <k, and S,(k,)<S,(k,) = i, Si, . We note that, for 
k<k, i =i, if  S,(k)<S,(k) and {mp*|a>1 and mp“ < S,(%)} = 
{mp*|a>1 and mp* < S,(k,)} 


where £,=0 for S,(k,)#mp", if S,(k,)=mp* then 


2. Proposition. [f p is a prime number and p25, then S,>S,, and S, > Sp 


Proof. Because p-1< p it results that 5, < S,. Of course p+ 1 is even and so: 
(i) if p+1=2', then i >2 and because 2i < 7 -1= p we have Sp, < Sp- 

if lz, 1 = p’ - pz... pr = = 8, = 
(ii) if p+1l#2', let p+1=p}!-pz---py, then S,,,(k) Bente ate 5 in (K) 


= Sp, (int). 





jm so < p it results that S (k) < S,(k) for k EN", so that 


Because p,,-i,, S 


S 


pei < Sp- 
3. Proposition. Let p,q be prime numbers and the sequences of functions 
(Sie? {Ss} sen" 
If p<q and i< j, then Sy <5). 


Proof. Evidently, if p<q and i< j, then for every k e N° 
Sy (K) < S y (k) < S (4) 
so, Sg <S 
4. Definition. Let p,q be prime numbers. We consider a function S J a Sequence of 
functions {S F Jen” and we note: 


i= max{i|S, S Sp} 
40 


i = min iS, <5}, 


then {k eN li <k< MY=A aise A , defines the interference zone of the function S, 5 


with the sequence is, 


5. Remarque. 
a) f Sy <S; for i eN'°, then nov exists jy and f= 1, and we say that Sy is separately 
of the sequence of functions {s 3} : 
P tieN? 


b) If there exist k EN” so that S, <S <S, , then A =@ and say that the 


F(a’) 


function S, does not interfere with the sequence of functions | 5 aa ee 
ieN 


6. Definition. The sequence {x,} w generaly increasing if 
VneN 3m EN" so that x,2x, for. m2m. 


7. Remarque. If the sequence {x,} „> with x,20 is generaly increasing and 
boundled, then every subsequence is generaly increasing and boundled. 


8. Proposition. The sequence ESaC E oye , where k e N°, is in generaly increasing 
and boundled. 


Proof. Because S,(k)=S5,(1), it results that {SalI ye is a subsequence of 
(SnD) pen" 
The sequence {S,,(1)} mey’ ÍS generaly increasing and boundled because: 


Ym eN’ 3t =m! so that Yt >f S,(1) 2 S, (1)=m25,(1). 


From the remarque 7 it results that the sequence {Salk} oye is generaly increasing 
boundled. 


9. Proposition. The sequence of functions {S,} | ew 5 generaly increasing boundled. 
Proof. Obviously, the zone of interference of the function S, with {S,} ev’ ÍS the set 
Amy = {k EN” {ry < k < n™} where 
m = max {n E N'|S, < S,} 


n™ = min {n € N'|S,, < S,}. 
at 


The interference zone Awm is nonemty because Sm € Amm and finite for S, < Sm S Sp» 
where p is one prime number greater than m. 
Because {5,(1)} is generaly increasing it results: 


VmeN’ 3t,¢N° sothat S,(1) 2 S„(1) for Vt 2%. 


For 7 =f, +n”) we have 


S,25,25S,(1) for Yr 2h, 


so that {S,} eN’ is generaly increasing boundled. 


10. Remarque. 
a) For n = p} - p}--- p? are posible the following cases: 


1) 3 k €{1,2,...,7}s0 that 
Sy SSy for j €{1,2,...,7}, 
then S, = Spp and pit is named the dominant factor for n. 
2) 3k,,ky,.--.¥q €{1,2,...,7} so that : 


Vtelm 3q, €N” sothat S,(g¢,)=S iz, (Ge) and 
Py, 


VieN S,D= efs, ol 


We shall name { pE 7 €1,m} the active factors, the others wold be name passive factors 
for n. 
b) We consider 
Npp = {n= pi - p3 li,i € N°}, where p, < p, are prime numbers. 
For n € Np, appear the following situations: 
1) į €(0,i¢2)], this means that pi is a pasive factor and p? is an active factor. 
2) i, € (iu) 4) this means that p;' and p? are active factors. 


3) i, €[i*® , ©) this means that pi is a active factor and p? is a pasive factor. 
42 


For p, < p, the repartion of exponents is represently in following scheme: 







The zone of exponents 
for numbers of type 1) 


wu Wwe 






for numbers of type 3) 


For numbers of type 2) 4 Elipi?) and i, € (iggy sf?) 


c) I consider that 


Namm = {n= Pt "P? -Pilih eN’}, 


where p, < p, < p, are prime numbers. 
Exist the following situations: l 
1) n e N”! ,j=1,2,3 this means that p/ is active factor. 


2) ne N”!™,jæk;, j,k €{1,2,3}, this means that pipe are active factors. 

3) ne NANPA this means that p!', p3, p} are active factors. N"”" is named the S- 
active cone for N a p ps- 

Obviously 


NARB = {n= p? P liii EN” and i, Elui) where j= k; j,k e{1,2,3}}. 


The repartision of exponents is represented in the following scheme: 


P,P 
NF 1F3 NPI] 





For p, < p, the repartion of exponents is represently in following scheme: 







The zone of exponents 
for numbers of type 1) 


wu Wwe 






for numbers of type 3) 


For numbers of type 2) 4 Elipi?) and i, € (iggy sf?) 


c) I consider that 


Namm = {n= Pt "P? -Pilih eN’}, 


where p, < p, < p, are prime numbers. 
Exist the following situations: l 
1) n e N”! ,j=1,2,3 this means that p/ is active factor. 


2) ne N”!™,jæk;, j,k €{1,2,3}, this means that pipe are active factors. 

3) ne NANPA this means that p!', p3, p} are active factors. N"”" is named the S- 
active cone for N a p ps- 

Obviously 


NARB = {n= p? P liii EN” and i, Elui) where j= k; j,k e{1,2,3}}. 


The repartision of exponents is represented in the following scheme: 


P,P 
NF 1F3 NPI] 








d) Generaly, I consider N, p p, = {n= Pi Be o -p lish,- EN }, where 
Pi <P, < -+ < p, are prime numbers. 
On NV, p.p, exist the following relation of equivalence: 


npm < n and m have the same active factors. 


This have the following clases: 

- N?^ , where j e{1,2,:--r}: 

neN’”^ on hase only p active factor 

- N?APR where j, # j and jo jz €{1,2,....7}. 
n e NPPPR e n has only pa l p? active factors. 


NPAP:?---Pr wich is named S-active cone. 


PIP2---Pr = 
N {neN Guach 


Obviously, if n e NPP2-# , then i, Eliapi”) withk+j and k,/ €{1,2,...,r}. 


n bas p} , p2 ,..., p7 active factors}. 


REFERENCES 
[1] I. Balacenoiu, Smarandache Numerical Functions, Smarandache 
Function Journal, Vol. 4-5, No.1, (1994), p.6-13. 


[2] I. Balacenoiu, V. Seleacu Some proprieties of Smarandache functions of the type I 
Smarandache Function Journal, Vol. 6, (1995). 


[3] P. Gronas A proof of the non-existence of "Samma". Smarandache 
Function Journal, Vol. 4-5, No.1, (1994), p.22-23. 


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