# Full text of "The Monotony of Smarandache Functions of First Kind"

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

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

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

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

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