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

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 ,(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 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.