on the calculus of smarandache function

8/12/2019 ON THE CALCULUS OF SMARANDACHE FUNCTION

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ON THE CALCULUS OF SMARANDACHE FUNCTIONby

C. Dumit.rescu ILDd C. RocaoreanuVniver3ity o Craiova,Dept. o Math.,Craiova I100),Rnrnaniu)

Introduction. The Smara.ndache function S : N - N is defined [5] by thecondition that. 5(n) is the smallest. integer m such tha.t m is divisible by n.So,wehave 5 1) = 1,5(212) = 16.

Considering on the set N two laticeal structures N = (N, /\, v) and Nd =(N, /\, ~ , w h e r e /\ = min, V = max, /\= the grea.tLcst comlllon divuJOr, ~ = t h ~d dsmallest. common multiple,it results that 5 halt the followings properties:d( 1) 5('1\ V 11) =S( 1l) V.5 ( 11)

~ 1 ) 1 11 ~ c i n1 S nd ~ S(71'J)where ~ iat the order in the lattice N and ~ c i i the order in t.he lattice N. 1t lsaid tbat

. n ~ n n dividE n zFrom these propertie1 we deduce that in fact on must c:olllli


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, " n)n = p} l )by Legendre's formula is said ~ h a L

e,(n) =L: [ ~ ]pAlso we have

( ) _ n - C1 p) n)e, n - p l (2)where [:r] is the integer pa.rt of x and C1(p){n) ill the Hum of digit.s of n ill thenumerical scale

P) : 1 , P I p l , ... , p' ...For ~ h e calculus of S(p ) we need to consider in ILdditiuu a gt llcraliaed lIuuu r

ical scale P] given by:(P]: tll(P) , tl:l(p) , ... , rl,(p) I

where Cli P) = (pi. - I)/ P - 1).Then in [3] it is showed thatS pQl) = p Ct{pj) p) (3)

that is the value of S pLl) i J obta.iued multiplyillg p by the number b t i n ~ d writingthe exponent Ct in the genera.li8ed lica.le P] and "rea.ding" iL in the U8Ua. IiCa. C (p).

Let us observe that the calculus in the gencra..l.iHed "calc P] ill C8HCutially different. from t.he calculus in t.he ecale (p).That. is becaulJe if we DoLe

then for t.he usua. 8ca. e P) it reaultll the rcclJ relll'C reiatiullb,,+l(P) = P . b,,(P)

and for the generalised acale [P] we have~ + l P ) = P . Clo (P) + 1

For Lhis,to add some numbers in the dcale P] we do as follows:

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1)Wesian\0 add from thedigits of decimals ,thatis from thecolumncorrespondingtoCll P).2) faddingsomedig it8 it isobtainedPfl' l(p ),t henwe uti li se anuntfrom the

daslfe of units (thecolumncorreHpondillg Lo LI p Lo ohLain p r.J p) 1=a3(p).Contiuuingtoadd,ifaga.iuH itisobt. a inedp tl -l (p ),th clI anew unitUJUHL1)(ueedfromtheclaaae ofuniLal,eLc.Ezampie. f

thenmn r= 442412

44dcba

Tofind thedigitsel, b,c, dwe sta.r t toaddfrom thecolumncorresponc iing toCl l(S):4Cll(5) S ) ~ 5 ) =Cl l(5) 4Cl l(5)

Now ,iIweta.ke a.n unitfromthef ir stcolumnweget:

1 b=4.CODiiDumgtheaddit. ionw have:

4a35) 4a3 5) a35)=a35) 4a35)andusinga. Dew unit(fromthefir stcolumn)itresu lt s:

4a3 (5) 4a35) a3S) 1=a.5) 4a35)1 c=4andd=1.

Finaly,addingtheremainedunits :4al(6 ) 21(5)=al5) 1(6)=5a l 6) 1=VJ5)itresult.thatthedigitb=4must becha.n gedinb=5II.nd l =OSo

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Replacing this expression of a in (6) we get.:(p- 1)' p - 1S(pa) = (C


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S, cr) = p cr - i, cr 12Aa il ia abowed in [4] we have 0 i,(O ) [0.;1 JThen it result.s that for each function S there exiHLH a function i, 1.10 that WI:have the linear combination

In [I} it is proved that~ S , c r ) + i, cr) =P ( 13)

(14)and 10 it is an evident analogy between the expression of e,(0 ) given by thtequality (2) and the expression of ip(o-) in (14).In [ ] it is also showed that

0 = (0([,1)(,) + _ (W(O )] = (O [,J)(,) + 0 - 0 [,1(0 )P P Pand 10

(15 )Finaly,let us obeerve that from the definition of Smarandache fUllction it resultstha.t

0-S, 0 e,)(o-) =p[-] = - - a,pwhere o p is the remainder of cr modulus p.Alao we have

(


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O (P}(Z) : O'lP (a) : O (p){X - 1) 1The calculus of card(S-l (n) ).Let. qll qll "'1 qlt be all t.he prime it.egera IIma.ll-

es1. 1.hen n a.nd non dividing n. LeI. also denoLe shortly e"/(n) = IrA soluLion Xoof 1.he equaUonS x) = n

has the property that. Xo divides n a.nd non divides (n - l) .Now, if d(n) s thenumber of p08it.ive divisors of n,from t.he inclusion

{m / m divides (n - I) } C {m / m divides n }a.nd using t.he definit.ion of Smara.ndache function it. results t.ha

card(S-l(n)) =d(n ) - dn - I) ) ( 16)Ezample. In [6] A. St.upuu a.nd D. W. Sharpe u proved t.haL it p ia a given

prime,t.be equauonS(=) =

has just d P -I) ) solutiolls (all of t.hem in bet.ween p aud p ) .Lt,t. us observe t.hate,(p ) =1 and e,p - I) ) = 0,80 because

d(P ) = (e,.(p ) 1)(11 1)(f2 1)...(/h 1) =2(f1 l)(h 1) ... f1t. 1)d((p -1) ) = f1 l)(h 1) ... 1)it results

card(S-l(P = (P ) - dP - 1) =dp - 1) )References1. M. Andrei , 1 Balacenoiu, C. DUUlit.rt'Hcu, E. RaJ 'lK:U, N. RadNK:u, V.

Seleacu, A Linear c O l U b i l l a ~ i o l l wit.h ~ m a m l l d u d l l FUlldlOIl to ubtlLll lrt t.lu' Idt lIt lLy,Proe. 2 6 ~ Annual iranian Math. C o n J e r e n c ~ , 1995},4.37-439.

2. M. Andrei, C. Dumit.reacu, V. Selea.cu, L. Tu Lescu 1 SL. Zanfir , La {ouction de Smarandache une nouvelle fonct.ion daus la Lheorie det nombrt H, C i ) T ~ J r l : 3International H. Poinca.re, Nancy, 14-18 May, 1994.

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on the calculus of smarandache function

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