SMARANDACHE NUMERI CAL FtJNCTI ONS by Ion Balacenoiu Dapar~arnan~ of Ma~hema~ics Univ.rsi~y of Craiova,Romania F.Smarandache defines [1) a numsricaL functton • 5 :~ --- ~ .SCn~ is the smaLLest inte~er m such that m! is diuisibLe by n. UsinS certain res1..J.L ts on standardised str1..J.ctures, three ~inds of Smarandache functions are defined and are etabLished some compatibiLity reLations between these /1..J.nctions. 1. Standardising functions.Le~ X be a nonvoid set, r c XxX an equ1valence rela~i~n, X ~he corresponding quo~i.n~ se~ and (I,SJ a ordered ~o~ally se~. 1.1 Definition.If 9 : X --- I is an arbi~rarely injec~ive func~ion, t.hen f X --- I defined by f(xJ = gCxJ is a s~andardising funct.ion. In t.his case t.he se~ X is said t.o be ( r , C I , SJ , f J s~andardised. = If rand r are ~wo equivalence rela~ions on X,~hen r r ~ r is 1 4t 1 4t defined as x r y if and only if x r y and x r y. Of course r 1 s 1 Z an equivalence rela~ion. In t.he following theorem we consider func~ions having ~he same x ___ I mcnc~on1cit.y.The func~ions f i = 1, S are of ~he same monc~onici~y if for every x,y from X i~ resul~s = f Ie C xJ S f Ie C yJ if and only if f.CxJSfCy) for k,j 1,s J J 6 x 1. 2 Theorem. I f the standardi si n9 funct.1. ons f -----+ I corresponding t.o t.he equ~valence relations r , i = 1, S , ar e of '- t.he some monot.onicit.y t.hen f = m41.X { f } is a st.andardi sl.ng '- '- funct.l.on corresponding to r = r ha.vl.ng same monot.onicl.t.y as f. '- = Proof. We give t.he proof of t.heorem in case s 2 .Let. x x r r 1 Z x be t.he equl.valence clases of x corresponding t.o r rand t.o 1 2 r = r r respect.l.vely and X X x t.he quotient. set.s on X. t 2 r r 1 2 We have f (x) = g (x ) and f (xJ :z 9 ex) , ....,her e 1 t r 2 2 r 1 2 gi X ---+. I , i=t,2 are inject.ive functions. The funct.ion r '- 9 X -->I defined by g(x )= ~~x{g ex ),g (x )} is inject.ive. r r 1 r 2 r t 2 Indeed if X- 1 ~ X- 2 and max{g (x- t ),g (x- t· )} = r r 1 r 2 r t 2 = - 2 - 2 max{g (x ),g (x)} ,then be cause of the injectivit.y of 1 r 2 r t 2 and ""'e ha ve for exampl e max{g (x"1 ), 9 (x"1 )} = t r 2 r 1 2 = 9 (x- 1 ) = 9 (x- 2 J = max {g (x- 2 J, 9 e x- 2 J ) and ....,e obt.a.in a t r 2 r t r 2 r 1 ;z t 2 cont.radict.ion because f (x2) = g(x- 2 J < 9 (x- t J = f (Xl) t t r t r t t 1 f (xl) = 9 (x"t J < g ex-2 J =f (xz) ,t.hat. is 2 2 r Z r 2 t z f and f are not. of t.he sa.me monot.onicity From the injec- 1 2 t.ivit.y of 9 it. result.s t.hat. f:X ---+.I defined by f( xJ = g( x J r is a st.andardising funct.ion.In a.ddit.ion we have g(x- I ) S g(x-2 ) <-+ max{ g (x- 1 J, 9 (x- 1 J ) S max{ go (x- Z J, 9 (x- 2 ) } ~ r r t r Z r t r Z r 1 2 1 2 1 t • 2 2 1 2 max{ f (x J , f eX )\.-S max{ f ex J. f (x J) ... f (x )Sf ex J and 1 2 .. l 1 2 1 1 f are of t.he same monot.onicit.y. Z 7 Le~ us supose now that T and ..L are two algebraic lows on X and I respecti vel y. 1. 3. Definition. The standardising function f: X __ I :.s said t.o :::e with T and ..L if for every X.y :.n X lhe t.r:.ple+_ C[CX), fCy), fCxTY)) satisfies ~he condilion ~. In lh:.s case it is said lhat. lhe function f -st.andardise t.he st.ruc- ~ t.rure CX'T) in t.he st.ructure CI,S,..L). * For example,if f is the Smarandache function S: IN --+ IN ,C SC n) :.s t.he smallest int.eger such t.hat. C SC n))! is di vi s:. bl e by n) lhen we get. t.he following ~-stadardisations: a) S ~ -standardise (IN * ,.) in (IN * ,S,+) because we have 1 :SCa.b)SSCa)+SCb) ~ 1 b) but. S verifie also lhe relat.ion !: • maxCSCa) ,SCb))SSCa. b) SSC a) . SCb) 2' • * . so S ~ -standardi se the structure C IN ,.) in CIN ,:S,.) 2 2. Smarandache functions of first kind •. The Smarandache function S is defined by means ot t.he tollowing functions S ~ for every prime number p let S: IN* -> IN* having p p n t.he property that CS Cn))! is di visibl e by p and is the p smallest. posit.ive integer with this property. Using the not.ion ot st.andardising functions in t.his section we give some generalisa- sion ot s. p 2.1.Definition.For every n IN* the relation reIN-x IN* is deti E n * ned as -follo......-s: i) if n= u1.c u=l or u2 p number pr i me • i EfN ) and a.b EO IN t.hen a r n b it and only it it exi st.s leE IN- such i.a l.b t.hat. Ie: ! :II M u Ie: ! = M U and le is the smallest posit.ive :.nt.eger with t.his propert.y. 8 I. ~ I. 1.1.) if :"1== PI 1 . Pz z P Ii t..hen a :- = r " r " " r n P t. 1 PzI. z P I. Ii 1 ~ 2.2. Daf i ni t. ion. For each n E t..he Smarandache funct..ion of . - first. kind is t..he numerical funct.ion S : IN -> IN def'i ned a.s follo ..... s n i) if n :2 ul.C u==l or u=p number pri me) t..hen S C a) = k, k bel:"lg n la t..he smallest. posit..ive int..eger ..... i t..h t..he propert..y t.hat.. lc:! = M u ... 1.. I. 1 2 s 1.1. ) 1. n = Pi . P2 P5 t..hen S n Ca) = l~t~; S P I. J C a.)} j Let.. us observe t..hat.. a) t..he funct..ions S are st..andardising funct.ions corresponding n • t..o t..he equivalence relat.ions r and for n=l ..... e get. x = IN - n r 1 for every xe IN and S Cn) = 1 for every n. 1 b) if n=p t.hen S is t..he funct.ion S def'ined by Smarandache. n p c) t..he funct.ions S are increasing and sO,are of' t.he same mono- n Lonicit..y in t..he sense given in t..he above sect.i on. . - 2.3.Theore~The f'unct.ions S ,for n @ IN , L -st.andardise CIN ,+) in n 1 * CIN ,::S,+) by L: max{S C a) ,S C b) }::SS C a+b)::SS C a) +S C b) for 1 n n n n n 1\.,* * * ,::s,.) every a, b E Ln and L z -st..andardise C IN ,+) in CIN by • ::s ::s L z max-CS n Ca) ,Sn Cb)} S n Ca+b) S n Ca). S n Cb) ,f'or every a,b E IN * P.ro of. Let., fo-r inst..ance, p be a prime number ,n=pi .,.1 E IN and a == S C a), b = S . C b) ,le" S C a+b) .Then by t.he def'init.ion of' S n I. I. I. P P P CDef'init.ion 2.2,) t.he numbers a * ,b - ,le are t.he smallest. posi- t..ive int.egers such t.hat. a•! =Mipa, b-c== u...p i.b and le!=MpiCa-+b). - Because a ::Sle and b * ::s le ,so max{a ,b )::s Ie: That.. is- -t..he firs_t. .in equalit..ies in Land L holds. 1 z = • b*) = .u..._.. ·,. b-!-- u... piCa+b) No ...... Ca +b)! a !Ca +1). . Ca + and 9 • • so k ~ a + b which implies that r: is valide. 1 1-• If p , fro. the first case we have • r: . 1Ia.x{S ( a) , S (b)} ~ 5 ( a+b) ~ 5 . (a) + S . (b),j=i7s t . P \. J P \. j p I. j pIJ. PIJ.. in consequence .ax{maxS (a) , aaxS (b)}~ aax{S . (a+b)} ~ 1l~{S (a)} + J pI.J J P i. J J P I. J. p.1-J J J J J = aax{ S (b) } , j 1,s That is J I- P J J Dlax{S (a),S (b)} S 5 (a+b) ~ 5 Ca) + 5 (b) n n n n n For the proof of the second part in r: let us notice that .... z (a+b) I ~ (ab) I ~ a+b ~ ab a >.1 and b > 1 and that ours inequality is satisfied for n=1 because 8 (a+b)=S (a)= 1 t = = 5 (b) 1. t • • Let now n>l.It results that for a = S (a) we have a > 1. Indeed, n • if then a = 1 if and only if S (a) = n. = = lIlax {s (a)} 1 =p = 1 J P I. J ' J • so n=l.It results that for every n>l we have S (a)= a >1 and n • • • • • Sn(b)= b > 1.Then (a +b )1 ~ (a .b )! we obtain 5 (a+b) S S (a) + S (b)~ S (a).S (b) froll n > 1. n n n n n • 3. SlIIarandache functions of ttl. second kind. For every n e IN , let S by the Smarandache function of the first kind defined above. n 3.1. Definition. The Smarandache functions of the second kind are • • k • the functions N ----+ IN defined by 5 (n) =8 (k), for k EfN • n We observe that for k=l the function SK is the snarandache function S defined in [lJ,with the modify S{l)= 1.Indeed for. = 11>1 sl(n) = Sn(l) Jn~X{Sp\(l)} = 1I~{Sp ( i. )} = sen). j J 10 3. 2. Theore& The 8marandache functions of the second kind L -stan- 3 • • dardise (~,.) in (~,~,+) by l:. max{l(a),l(b)} s sle(a.b) s slc(a) + Slc(b),for every a,b E~· 3 . • • and l: -standardise (IN ,.) in (~ ,~,.) by 4 Ie: Ie Ie le le • l: . max{S (a),S (b)} ~ s (a.b) ~ S (a).S (b),for every a,bEN 4· le Proof.The equivalence relation corresponding to sle is r, defined Ie • • • Ie by arb if and only if there exists a E IN such that a I =Ma , • • a 1= ~le and a is the smallest integer with this nu prop~rty. That is, the functions sle are standardising functions attached to le the equivalence relations r. This functions are not of the so.e monotonicity because, for exaa z z pIe, sZ(a) s SZ(b) ~ s(a ) s S(b ) and fro. these inequalities S1 ( a) s S1 (b) does not result. Jc • le • Ie Now for every a,b E IN· let S (a) = a ,S (b) = b ,8 (a.b)= s. • • Then a ,b,s are respectively these smallest positive integers • Ic • • Ie Jc Ie such that a = Ma , b I = Kb ,sl = M(a b ) and so sl =Ma = le • • • • =Mb ,that is, aSs and b S s,which implies that .ax{a,b }Ss lc Ic Ie or max { S (a), 8 (b)} S 8 (a. b ) (3.1) •• = •• = Ic Ie Because of the fact that (a + b ) 1 M (a 1 b I) M (a b ), it re- • • suIts that s ~ a + b ,so l ( a . b ) S sle ( a ) + Sic ( b ) ( 3 .2) From (3.1) and (3.2) it results that max{SI e (a),8l eb( )} -< S le(a ) + Sle(b) (3.3) Which is the relation ~ 3. • • •• Ie Ie Ie Fro", (a b ) 1 = M(a !.b !) it results that S (a.b) S S (a).S (b) and thus the relation ~ 4 11 4. The Snar~nd~che ftmctions of the third kind. We considere two arbitrary sequances (a) 1=a ,a '· .. ,a 1 z , ••• n (b) l=b,b, ... ,b .. 1 Z n b..:. wi th the properties that ~r"\ = ~. an' ~n = b • Obviously, there are n infinitely many such sequences;because chosinq an arbitrary value for a, the next terms in the net can be easily determined by the Z imposed condition. - b b - = Let now the function f :{N ->IN defined by f (n) 5 (b ), a- a- a- n n 5 is the 5marandache function of the first kind. Then it is ea- a n sily to see that : = = = ( i) for a 1 and b n n !N- it results that fb 5 @ n n' a- 1 (ii) for a = n and b = 1 n e {N- it results that fb = 51 n n' c:L 4. 1. Definition. The" 5marandache functions of the third kind are the b = fb functions S in the case that the sequances (a) and (b) a- a- are different from those concerned in the situation (i) and (ii) from above. - 4.2-.Theore~ The functions I:~-standardise (IN ,.) in (IN ,S, +,.) by I: : max {fb ( k) , fb ( n)} S fb ( k. n) S b . fb ( k) + h f~ (n) ~ a- c:L c:L n a- ~ _ Proof.Let = k-, f b (n) = 5 (b) = n • and c:L a- n n =5 (h) = t . Then .k -,n- and t are the smallest positive in- a- ~,.., len ~ • bn a.:n~r "\ teqers such that k I = H ~ , n 1= H a and tJ= H n of course, • • aax{k ,n } S t (4.1) 12 • • b • • b Now, because (bJ,: . n ) I = H(n I} Ie (b . k )! = H(k ! ) n and n - - - - - • ble b (bJ,: n + b k ) I = H[ (b n )!.Cb k )I] = H[ (n ! ) . (k ! ) n] = n lc n b blc b n = H[ (a (~ ) n ] it results that n - - t ~ bnk + ~n (4.2) Fro. (4.1) and (4.2) we obtain ( 4 .3) From (4.3) we qet L ,so the Smarandache functions of the ~ third kind satisfy L: max{S b (k),S b (n)} ~ Sb (1m) ~ b Sb (k) + h Sb (nl,for evry k,ne!N - " a. a. a. no. Ko. 4. 3. Example. Let the sequances Ca) and (n) defined by a = b = n. n n • neiN • The correspondinq Smarandache function of the third kind is S a. :rN----+rN • Sa . (n) = S Cn) and a: a. n - ,for every k,n&!N This relation is equivalent with the following relation written by meens th the SJftarandache function: '~i References [1] F.Smarandache,A Function in the Number Theory,An.Univ. Timisoara,seria st. mat Vol. XVIII,fasc.1,pp.79-88.1980. [2] Smarandache Function-Journal-Vol.l No.1,DeceMber 1990. 13