On some numerical functions

8/12/2019 On some numerical functions

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SM R ND CHE NUMERI C L FtJNCTI ONSb y

Io n B a l a c e n o i uD a p a r ~ a r n a n ~ o f M a ~ h e m a ~ i c s

U n i v . r s i ~ y o f C r a io v a ,R o man ia

F Smarandache d e f i n e s [ 1 ) a numsr icaL f u n c t t o n5 : ~ ~ . S C n ~ i s the smaLLest i n t e ~ e r m such t h a tm s d i u i s i b L e by n U s i n S c e r t a i n res1..J.L t s o n

s t a n d a r d i s e d str1. .J.ctures t h r e e ~ i n d s o f Smarandachef u n c t i o n s a re d e f i n e d and a re e t a b L i s h e d somec o m p a t i b i L i t y r e L a t i o n s b e tw e e n t h e s e /1..J.nctions.

1 . S t a n d a r d i s i n g f u n c t i o n s L e ~ X be a n o n v o i d s e t , r c XxX ane q u 1 v a l e n c e r e l a ~ i ~ n X ~ h e c o r r e s p o n d i n g q u o ~ i . n ~ s e ~ and I , S J a~ o ~ a l l y o r d e r e d s e ~ .1 . 1 D e f i n i t i o n . I f 9 : X --- I is an a r b i ~ r a r e l y i n j e ~ i v e f u n c ~ i o nt hen f X --- I d e f i n e d by f x J = gCxJ is a s ~ a n d a r d i s i n g func t . ion .In t .h is c a s e t he s e ~ X is s a i d t o b e r , C I SJ f J s ~ a n d a r d i s e dIf rand r a r e ~ w o e q u i v a l e n c e r e l a ~ i o n s on X ~ h e n r = r ~ r is1 t 1 td e f i n e d a s x r y if and o n l y if x r y and x r y1 Z f c o u r s e r 1 san e q u i v a l e n c e r e l a ~ i o n .

In t he f o l l o w i n g t heorem we c o n s i d e r f u n c ~ i o n s h av in g ~ h e samem c n c ~ o n 1 c i t y T h e f u n c ~ i o n s f x ___ I i = 1 S a r e o f ~ h esame m o n c ~ o n i c i ~ y if fo r e v e r y x , y f rom X i ~ r e s u l ~ s

f eCxJ S f eC yJ if and o n l y if f . C x J S f C y )J J f o r k j = 1 s

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1 . 2 Theorem. I f t h e s t a n d a r d i s i n9 funct.1. o n s f x ----- Ic o r r e s p o n d i n g t.o t .he e q u ~ v l e n c e r e l a t i o n s r , = 1 , S , a r e o f

-t .he some monot .onici t .y t .hen f = m41.X { f } s a s t . a n d a rd i s l .ng

- -

funct . l .on c o r r e s p o n d i n g to r = r ha.vl.ng samemonot.onicl . t .y a s f.

-

P ro o f . We g i v e t .he p ro o f o f t .heorem in c a s e s = 2 .Let. x xr r1 Zx be t .he equ l .va lence c l a s e s o f x c o r r e s p o n d i n g t.o r 1 rand t.o2r = r t r r e s p ec t . l . v e ly and X2 r X x t .he quo t i en t . se t . s on X.1

We have f (x) = g (x1 t r and1

r 2

f (xJ :z 9 ex , ....,her e2 2 r 2

g i X - - -+ . I , i = t , 2 a r e i n j e c t . i v e f u n c t i o n s . The func t . i onr-

9 X -->I d e f i n e d by g x = ~ ~ x { g e x ),g (x )} s i n j ec t . i ve .r 1 r 2 rt 2In d e e d i - 1 - 2X Xr r and - t - tmax{g x ),g (x ) }1 r 2 r =t 2

- 2 - 2= max{g (x ,g x ) }1 r 2 r , t h e n be c a u s e o f t h e i n j e c t i v i t . y o fand

- 1= 9 x )t r 1

t 2'e ha ve fo r exampl e 1 1max{g (x ), 9 x }t r 2 r

- 2= 9 x J2 r z1 2

- 2 - 2= max {g (x J , 9 e x J andt r 2 r ....,e obt.a . int 2- t

=a

con t . rad ic t . i on b ecau s e f x 2 - 2 < J f Xl)= g x J 9 t (x =

f 1 and f a r e not. o f2

tf x l )2

t .he

t r r tt 1t -2 =f x z )= 9 x J < g e x J , t .hat.r Z r 2t z

sa.me monot .on ic i ty From t h e i n j e c -is

t . iv i t .y o f 9 it r e su l t . s t.hat. f :X ---+.I d e f i n e d b y f xJ = g x Js a s t . a n d a r d i s i n g f u n c t . i o n . I n a.ddi t . ion we have

- Ig x r-2S g x

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e ~ us su p o se now t h a t T and L a r e two a lg e b ra i c lowson X and I r e s p e c t i ve l y.1 . 3. D e f in i t i o n . The s t a n d a r d i s i n g f u n c t i o n f : X __ I : .s s a i dt o e wi th T and L if f o r e v e r y X.y : .nX l h e t.r:.ple+_ C[CX), fCy) , fCxTY)) satisfies ~ h e c o n d i l i o n Inlh : . s c a s e it is s a i d lha t . l h e f u n c t i o n f ~ - s t . anda rd i se t .he s t . ruc-t . ru re CX'T) i n t .he s t . ruc tu re CI,S,. .L).

*or e x a m p l e , i f f is t h e Smarandache f u n c t i o n S: IN --+ IN ,C SC n): .s t .he s m a l l e s t i n t . ege r such t.hat. CSC n ) ) is d i vi s:. bl e by n) l henwe get. t .he f o l l o w i n g ~ s t a d a r d i s a t i o n s :

* *) S ~ - s t a n d a r d i s e1 IN ,. i n IN , S , + ) b e c a u se we have

b) but.

so S

~ :SCa .b)SSCa)+SCb)1S v e r i f i e a l s o l he r e l a t . i on

: maxCSCa) ,SCb))SSC a. b) SSC a) . SCb)2~ - s t a n d a r d i s e2 t h e s t r u c t u r e C N ,. in * .CIN , : S , . )

2. Smarandache f u n c t i o n s o f first k i n d . The Smarandachef u n c t i o n S is d e f i n e d by means o t t .he t o l l o w i n gf u n c t i o n s S f o r e v e r y pr ime number p let S: IN -> IN hav ingp pt .he p r o p e r t y t h a t is d i v i s i b l e by np and is t h eS C n ) )psmal l es t . pos i t . i ve i n t e g e r w i t h t h i s p r o p e r t y . Using t h e not . ion o ts t . anda rd i s i ng f u n c t i o n s in t .h is s e c t i o n we g i v e some g e n e r a l i s a -s i o n o t sp2 . 1 . D e f i n i t i o n . F o r e v e r y n E IN t h e r e l a t i o n r e I N - x IN i s d e t i nned a s follo......-s: i) if n= u1.c u=l o r u 2 p number *r i me i fN ) and- IN-a . b EO IN t .hen a r b it and o n l y it it e x i s t .s leE suchnt.hat. Ie M i.a Ie M l.b and le is t h e s m a l l e s t pos i t . iveII u = U: .nt .eger wi th t . h i s proper t .y .

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I. I1.1.) i 1 z i t..hen1== PI . Pz Pa

:- = r r rn t. I IP 1 Pz z P i12 . 2 . Daf ni . i o n . For e a c h n E t..he Sm a r a nda c he funct . . ion o f

. -i r s t . k i n d is t..he n u m e r i c a l fu n c t . i o n S : IN > IN def ' i ned a.s f o l l o sni) if n :2 ul.C u==l o r u=p number p r i me) t..hen S C a ) = k , k bel:"lgnt..he s m a l l e s t . p o s i t . . i v e int . .eger i t..h t..he proper t . . y t.hat.. lc: M la= u

1 I1 2 s1.1. .... n = Pi . P2 P5 t. .hen S Ca) = l ~ t ~ S I Ca . }n P JjLet.. us o b s e r v e t..hat..a) t..he func t . . i ons S a r en s t . . a n d a r d i s i n g f u n c t . i o n s c o r r e s p o n d i n g

t..o t..he e q u i v a l e n c e r e l a t . i o n sf o r e v e r y -e IN a nd S Cn) =1 r a n d f o rn1 f o r e v e r y n= l e get . x = INr 1n.b) if n=p t .hen S is t..he fu n c t . i o n S d e f ' i n ed b y Smarandache .n pc ) t..he fu n c t . i o n s S a r e i n c r e a s i n g a nd s O , a r e of' t .he same mono-nLonic i t . .y i n t..he s e n s e g i v e n i n t..he a b o v e sec t . i on.

- .3 T h e o r e ~ T h e f 'unc t . ions S , f o r n @ IN , L - s t . a n d a r d i s e CIN ,+ ) i n*IN ,::S,+) b ye v e r y b 1\.,*a , E Ln

L:1and

n 1max{S Ca ) ,S Cb ) }::SS Ca+b)::SS Ca ) +S Cb)n n n n n

L - s t . . an d a rd i s ez *IN , + ) i n *IN ,::s,.f o rby

max-CS Ca) ,S Cb)} s S Ca+b) s S Ca) . S Cb) , f ' o r e v e r y a , b E INz n n n n nPro o f . Let . , f o r i ns t . . ance , p be a p r i m e number i *, n=p ,1 E IN and. - == S Ca), b = S .Cb) , le" S Ca +b) . T h e n b y t .he d e f ' i n i t . i o n of'

I I IP P P* -

S nCDef ' in i t . i on 2 . 2 , ) t .he numbers a , b ,le a r e t .he s ma l l e s t . p o s i -t . . ive i n t . e g e r s s u c h t .hat.Be c a use

ia b - upi.ba =Mp, c== a nd le =MpiCa+b)- *::Sle a nd b s le , s o - .ax{a ,b )::s IeThat.. is t..he f i r s t . i n e q u a l i t . . i e s i n Land L h o l d s .1 z- - _.o ..... Ca +b) = a C a +1) . C b*) u _ , b- - - up iC a +b). a + = . a nd

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so k ~ a + b which implies1

tha t r: 1 i s val ide .I f p , f ro . the f i r s t case we haver: .t . 1Ia.x{S a , S\. \. b ) } ~ 5 ( a+b)I. ~ 5 . a ) + S . b) , j= i7sI. I . .P J P j p j pJ PJin consequence.ax{maxSJ I. (a) , aaxSJ i b ) } ~ aax{S . (a+b)} ~J I. . 1 l ~ { S 1p.JJ

(a)} +

aax{ SJ

p JJI-P JJ

b) }P JJ

, j = 1 sP JJ

That i sDlax{S a) ,S (b)} S 5 (a+b) ~ 5 Ca) + 5 (b)n n n n n

For the proof of the second pa r t in r:z l e t us not ice t ha t(a+b) I ~ (ab) I a+b ~ ab a >.1 and b > 1 and tha tours i nequa l i ty i s sa t i s f i e d= 5 b ) = 1 . for n=1 because 8 (a+b)=S (a)=1 ttLet now n > l . I t r esu l t s t ha t for a = S (a) we have a > 1 . Indeed,ni f then a = 1 if and only if S a ) =n= lIlax {s I. a ) } = 1J P J =p = 1.so

J

n=l . I t r e s u l t s t ha t fo r every n>l we have S (a)= a >nand

Sn(b)= b > 1.Then (a +b )1 ~ (a .b ) we obta in5 (a+b) S S (a) + S b ) ~ S a) .S (b) froll n > 1 .n n n n n

3. Sl I I a r andache f u n c t i o n s o f ttl s e c o n d k i n d . For every n e IN , l e tS by the Smarandache funct ion of the f i r s t kind def ined above.n3 . 1 . D e f i n i t i o n . The Smarandache funct ions of the second kind arethe func t ions - - - -+ IN def ined by k 5 (n) =8 k) , fo r k EfN nWe observe tha t fo r k=l the funct ion SK i s the snarandachefunct ion S def ined in [ l J ,wi th the modify S{l)= 1. Indeed for .> s l n ) = Sn(l) = n ~ X { S p \ l ) } = 1 I ~ { S p ( i j } = sen) .J

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3. 2 . Th eo r e& The 8marandache funct ions of the second kind L - s t an -3 dardise ~ , . ) in ~ , ~ , + ) byl : m a x { l ( a ) , l( b ) } s sle(a.b) s slc(a) + Slc(b),for every a ,b E ~ 3 .andl : .4

l : - s t anda rd i se4 IN ,. in ~ , ~ , . ) byIe e e le le max{S (a ) ,S ( b ) } ~ s (a .b) ~ S (a ) .S (b) , for every a,bEN

Proof.The equivalence re l a t ion cor responding to sle i s ler, def inedby e earb if and only if there e x i s t s a E N such t h a t a I =Ma ,

~ l a = nu and a i s the smal l e s t i n t ege r with t h i s p r o p ~ r t y .That i s , the func t ions sle are s t andard i s ing funct ions a t tached tothe equivalence r e l a t i o n s

ler.

This func t ions a re not o f the so .e monotonici ty because, fo r exaa-pIe , sZ(a) s SZ(b) s (az ) s S(bz ) and f ro . these inequa l i t i e sS a) s S (b) does not r e s u l t .Now fo r every a ,b E IN l e t Jc le eS (a) = a ,S (b ) = b ,8 (a .b)= s . Then a ,b,s are respec t ive ly t hese sma l l e s t p o s i t i v e i n t ege rs

c e Jc esuch t ha t a = Ma , b I = Kb , s l = M(a b ) and so s l =Ma =le =Mb , t h a t i s , aSs and b S s ,which impl ies t h a t . a x { a , b }Ss

or lc c emax { S (a ) , 8 ( b ) } S 8 (a. b ) (3.1)Because of the f ac t t h a t c e(a + b ) 1 = M(a 1 b I ) = M(a b ), it re - su I t s t h a t s ~ a + b ,so

. b ) S sle a ) + Sic b ) 3 .2)From (3.1) and (3.2) it re su l t s t ha t

e l e ( ) } < le ) Sle b)max{S ( a ) ,8 b - S a + (3.3)Which i s t he r e l a t i o n ~ 3.

Fro , e e e(a b ) 1 = M(a .b ) it r e s u l t s t h a t S (a .b) S S (a ) .S b)and thus the r e l a t i o n ~ 4

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4 . The S n a r ~ n d ~ c h e f t m c t i o n s o f t h e third k in d .We cons idere two a rb i t r a ry sequances a)

b )1=a1 , az an ,l = b , b , b ..1 Z n

w i th the prope r t i e s t h a t ~ r \ = ~ . an ~ n = b : bn Obviously, there arei n f i n i t e l y many such sequences;because chosinq an a rb i t r a ry valuefor a, the next terms in the net can be eas i ly determined by theZimposed condi t ion .

b - - bLet now the funct ion f :{N - > I Na def ined by f n) = 5 b ) ,a a nn5 i s the 5marandache func t ion o f the f i r s t kind. Then it i s ea-a- ns i l y to see t h a t :

( i) fo r a = 1 and b = n n @ N-n n it r sul ts t h a t fb = 5a 1i i ) fo r a = n and b = 1 n e {N -n n it r sul ts t ha t fb = 51c:L

4 . 1 . D e f i n i t i o n . The 5marandache funct ions o f the t h i r d kind are thefunct ions b fbS = a in the case t h a t the sequances a ) and (b)are d i f f e r e n t from those concerned in the s i t u a t i o n i ) and i i )from above.

4 2 T h e o r e ~ The funct ions I : ~ s t a n d a r d i s e -IN ,. in-IN ,S , + , . ) byI: : max {fb (k) , fb ( n)} S fb ( k . n) S b . fb ( k) + h (n)a c:L c:L n a _

Proof .Let- b = k, f n ) = 5 (b) = nc:L a n andn

=5 h) = t . Thena ~ . . - -,n and t are the smal l e s t pos i t ive in -lent eqe rs such t h a t . ~k I = H ~ ,

o f course , aax{k ,n } S t

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bnn 1= H an and ~ r \t J= H a :n

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On some numerical functions

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