SMARANDACHE TYPE FUNCTION OBTAINED BY DUALITY

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1 SMARANDACHE TYPE FUNCTION OBTAINED BY DUALITY C Dumtrescu, N Vîrlan, Şt Zamfr, E Răescu, N Răescu, FSmaranache Deartment of Mathematcs, Unversty of Craova, Romana Abstract In ths aer we extene the Smaranache functon from the set N of ostve ntegers to the set Q of ratonal numbers Usng the nverson formula, ths functon s also regare as a generatng functon We ut n evence a roceure to construct a (numercal) functon startng from a gven functon n two artcular cases Also connectons between the Smaranache functon an Euler s totent functon as wth Remann s zeta functon are establshe Introucton The Smaranache functon [3] s a numercal functon Sn = mn mm! s vsble by n { } * * S :N N efne by From the efnton t results that f n = t t () s the ecomoston of n nto rmes, then S(n) = max S( ) () an moreover, f [n,n ] s the smallest common multle of n an n, then S( [ n, n] ) = max { S( n), S( n) } (3) The Smaranache functon characterzes the rme n the sense that a ostve nteger 4 s rme f an only f t s a fxe ont of S From Legenre s formula: m m! = (4) t results [] that f numercal scale a = n n ( ) ( ) an b n (b) = n, then conserng the stanar [ ]:b (),b (),,b n (), we have [ ]: a, a,, a ( ), n S( )= ( [ ] ) (5)

2 that s S( ) s calculate multlyng by the number obtane wrtng the exonent n the generalze scale [ ] an reang t n the stanar scale () Let us observe that the calculus n the generalze scale [ ] s essentally fferent from the calculus n the usual scale(), because the usual relatonsh b n+ () = b n () s mofe n a n+ () = a n () + (for more etals see []) Let us note from now on S () = S( ) In [3] t s rove that S () = ( ) + σ [ ] () (6) where σ [ ] () s the sum of the gts of wrtten n the scale [ ], an also that ( ) S ( ) ( E ) [ ] = + + σ + σ (7) where σ () s the sum of the gts of wrtten n the stanar scale () an E () s the exonent of n the ecomoston nto rmes of! From (4) t results that E () =, where [h] s the ntegral art of h It s also sa [] that σ E = (8) We can observe that ths equalty may be wrtten as E () = [ ] that s, the exonent of n the ecomoston nto rmes of! s obtane wrtng the ntegral art of / n the base () an reang n the scale [ ] Fnally, we note that n [] t s rove that S () = + σ [ ]() (9) From the efnton of S t results that S( E ) = = ( s the remaner of wth resect to the moulus m ) an also that E( S ) ; E( S ) < () so S σ ( S ) S σ ( S ) ; < Usng (6) we obtan that S () s the unque soluton of the system σ (x) σ [ ] () σ (x ) + () Connectons wth classcal numercal functons It s nown that Remann s zeta functon s

3 ζ(s) = n s n We may establsh a connecton between the functon S an Remann s functon as follows: Prooston If nteger n then tn n= s the ecomoston nto rmes of the ostve ζ ( s ) ζ () s = n tn = ( n ) S s n n = Proof We frst establsh a connecton wth Euler s totent functonϕ Let us observe that, for, = ( -) a +, so σ [ ] ( ) = Then by usng (6) t results (for ) that S ( ) = ( ) + σ [ ] ( ) = ϕ( ) + Usng the well nown relaton between ϕ an ζ gven by ζ(s ) ϕ(n) = ζ(s) n n n an (), t results the requre relaton Let us remar also that, f n s gven by (), then an t t ϕ ( ) ϕ( n) = ( ) = S ( ) = = ( ) + ϕ Sn = max ( ) + Now t s nown that + ϕ( ) + + ϕ( ) = an then S ( ) = = Consequently we may wrte Sn = max S S( ) ( ) = To establsh a connecton wth Mangolt s functon let us note = mn, = max, We shall wrte also n n = ( n, n ) an n n = [ n, n ] L = the greatest common vsor, an = the smallest common multle The Smaranache functon S may be regare as functon from the lattce * N,, * =, nto lattce L = N,, such that () S n = S( n ) =, =, 3

4 Of course S s also orer reservng n the sense that n n S(n ) < S(n ) It s nown from [] that f ( V,, ) s a fnte lattce, V = { x, x,, xn} wth the nuce orer, then for every functon f : V N the assocate generatng functon s efne by F(x) = f (y) (3) y x Mangolt s functon Λ s ln f n = Λ(n) = otherwse The generatng functon of Λ n the lattce L s F ( n) = Λ = lnn (4) n The last equalty follows from the fact that n n = \ n ( ves n) The generatng functon of Λ n the lattce L s the functon Ψ F(n) = Λ() =Ψ(n) = ln[,,,n] (5) n Then we have the agram from below We observe that the efnton of S s n a close connecton wth the equaltes () an () n ths agram If we note the Mangolt s functon by f then the relatons [,,,n] = e F(n) = e f () e f () e f (n) = e Ψ(n) F% F () F () F ( n) n! = e = e e e together wth the efnton of S, suggest us to conser numercal functons of the form: ν ( n) = mn { m/ n [,,, m] } (6) whch wll be etale n secton 5 4

5 Λ L L F (n) = /n Λ() = lnn () () F(n) = n Λ() =Ψ(n) L L L L %F = ln /n () () (3) (4) %F(n) = ln = lnn! Ψ % = Ψ() n /n %Ψ(n) = Ψ() n 5

6 3 The Smaranache functon as generatng functon Let V be a artal orer set A functon f : V N may be obtane from ts generatng functon F, efne as n (5), by the nverson formula f (x) = F(z)μ(z, x) (7) z x where μ s Moebus functon on V, that s μ : VXV N satsfes: (μ )μ(x, y) =, f x y (μ )μ(x, x) = (μ 3 ) μ(x, y) =, f x < z x y z It s nown from [] that f V {,,, n} = then for V, we have μ(x, y) = μ y x where μ() s the numercal Moebus functon μ() =, μ() = ( ) f = an μ() = f s vsble by the square of an nteger > If f s the Smaranache functon t results F S (n) = S(n) /n Untl now t s not nown a close formula for F S, but n [8] t s rove that () F S (n) = n f an only f n s rme, n = 9, n = 6, or n = 4 () F S (n) > n f an only f n { 8,, 8, } or n = wth a rme (hence t results F S (n) n + 4 for every ostve nteger n ) an n [] t s shown that t () F( t ) = = In ths secton we shall conser the Smaranache functon as a generatng functon, that s usng the nverson formula; we shall construct the functon s such that s(n) = μ()s n (8) /n If n s gven by () t results that n s(n) = ( ) r S r r Let us conser S(n) = max S( ) = S( ) We stngush the followng cases: (a ) f S( ) S( ) for all then we observe that the vsors for whch μ() are of the form = or = r A vsor of the last form may contan or not, so usng () t results t t t t ( ) t t t t t t t or S( ) an s(n) = otherwse sn = S C + C + + ( ) C + S + C C + + ( ) C that s s(n) = f, 6

7 j (a ) f there exsts j such that S( ) S( ) j j S( j ) S( ) for, j j j j we also suose that S( j ) max S( )/ j S S j < an { } = < Then t t j sn = S C + C + ( ) C + S + C C t t + ( ) C + ( )( t t t ) ( j )( t t t ) j ( )( t t j ) t t t + S C + C + C j j so s(n) = f t 3 or S( j ) S( j ) = an s(n) = j otherwse Consequently, to obtan s(n) we construct as above a maxmal sequence,,,, such that S( n) = S( ), S( ) < S( ),, S( ) < S( ) an t results that s(n) = f t + or S( ) = S( ) an s(n) = ( ) + otherwse Let us observe that S( )= S( ) ( ) + σ [ ] () = ( )( ) + σ [ ] ( ) σ [ ] ( ) σ [ ] () = Otherwse we have σ [ ] ( ) σ [ ] () = So we may wrte s(n) = f t + or σ [ ]( ) σ [ ] ( ) = ( ) + otherwse Alcaton It s nown from [] that ( V,, ) s a fnte lattce, wth the nuce orer an for the functon f : V N we conser the generatng functon F efne as n (5) then f g j = F x x j t results et g j = f (x ) f (x ) f (x n ) In [] t s shown also that ths asserton may be generalze for artal orere set by efnng g j = f (x) x x x x j Usng these results, f we enote by (, j) the greatest common vsor of an j, an Δ () r = et ( S( (, j) )) for, j =, r then Δ(r) = s() s() s(r) That s for a suffcent large r we have Δ(r) = (n fact for r 8 ) Moreover, for every n there Δ ( nr, ) = et S n+ n, + j =, for, j=, r exsts a suffcent large r such that because Δ(n,r) = Sn+ n = 7

8 4 The extenson of S to the ratonal numbers To obtan ths extenson we shall efne frst a ual functon of the Smaranache functon In [4] an [6] a ualty rncle s use to obtan, startng from a gven lattce on the unt nterval, other lattces on the same set The results are use to roose a efnton of b-toologcal saces an to ntrouce a new ont of vew for stuyng the fuzzy sets In [5] the metho to obtan new lattces on the unt nterval s generalze for an arbtrary lattce Here we aot a metho from [5] to construct all the functons te n a certan sense by ualty to the Smaranache functon ( n) = m/ n m! L ( n) = m/ m! n, Le us observe that f we note R { }, { } R ( n) = { m/ n m! }, ( n) = { m/ m! n} by the trlet ( R,, ), because Sn = { m/ m ( n) } L then we may say that the functon S s efne R Now we may nvestgate all the functons efne by means of a trlet ( a,b,c), where a s one of the symbols,,,, b s one of the symbols an, an c s one of the sets R (n),l (n),r(n),l(n) efne above Not all of these functons are non-trval As we have alreay seen the trlet (,,R ) efne the functon S (n) = S(n), but the trlet (,, L ) efnes the functon S n = { m m n}, whch s entcally one /! Many of the functons obtane by ths metho are ste functons For nstance let S 3 be the functon efne by (,,R ) We have S ( n) = 3 { m/ n m! }, so S 3 (n) = m f an only f n ( m! ) +, m! Let us focus the attenton on the functon efne by ( L,, ) { } S (4) /! 4 = m m n (9) where there s, n a certan sense, the ual of Smaranache functon Prooston 4 The functon S 4 satsfes ( = ) = * * so t s a morhsm from ( N, ) to () S n n S ( n ) S ( n ) N, Proof Let us enote by,,,,the sequence of the rme numbers an let n = β, n = mn( The n n =,β ) If S4( n n) = m, S 4 (n ) = m, for =, an we suose m m then the rght han n () s m m = m By the efnton S 4 we E ( m) mn, β for an there exsts j such that have 8

9 E (m + ) > mn(,β ) Then > E ( m) an β E (m) for all We also have E (m r ) for r =, In aton there exst h an such that E h (m + ) > h, e j (m + ) > Then ( β ) ( ε ε ) mn, mn ( m ), ( m ) = E ( m ), because m m, so m m If we assume m < m t results that m! n, therefore t exsts h for whch E h (m) > h an we have the contracton E ( m) mn {, } h h βh S4 ( n+ ) = an > Of course S 4 (n) > f an only f n s even () Prooston 4 Let,,,, be the sequence of all consecutve rmes an n = β q β q βr q r * the ecomoston of n N nto rmes such that the frst art of the ecomoston contans the (eventually) consecutve rmes, an let S( ) f E ( S( )) > t = () S( ) + f E ( S( )) = then Sn = mn { t, t,, t, + } E S >, then from the efnton of the functon S results that ( ) S Proof If s the greatest ostve nteger m such that E (m) Also f E S = then S( ) + s the greatest nteger m wth the roerty that E (m) = It results that mn { t, t,, t, + } s the greatest nteger m such that E (m!), for =,,, Prooston 43 The functon S 4 satsfes S4( ( n+ n) ) S4( [ n, n] ) = S4( n) S4( n) for all ostve ntegers n an n Proof The equalty results usng () from the fact that ( n+ n, [ n, n] ) = (( n, n) ) We ont out now some morhsm roertes of the functons efne by a trlet a,b,c as above Prooston 44 () The functon S, S n = { m m n } :N * N * 5 /! 5 satsfes (3) S n n = S ( n ) S ( n ) = S ( n ) S ( n )

10 () () Proof The functon The functon S, S n = { m n m } :N * N * 6 /! 6 satsfes S 6 n n = S (n 6 ) S 6 (n ) (4) S :N * N *, { } 7 S /! 7 n = m m n satsfes ( ) =, S ( n n ) = S ( n ) S ( n ) (5) S n n S n S n () Let A { a / a! n } B bj / bj! n C c / c! n n Then we have A B or B A Inee, let A= { a a a }, B { b b b } =, { } =, an { } =,,, h =,,, r such that a < a + an b j < b j + Then f a h < b r t results that a b r for =, h so a! b r! n That means A B Analogously, f br ah t results B A Of course we have C = A B so f A B t results { } S n n = c = a = S ( n ) = mn S ( n ), S ( n ) = S ( n ) S ( n ) From (5) t results that S 5 s orer reservng n L (but not n L, because m! < m!+ but S 5 (m!) = [,,,m] an S 5 (m!+ ) =, because m!+ s o) () Let us observe that S6 ( n) = { m/, t such that E ( m) < } If a = { m/ n m! } then n (a + )! an a+ = { m/ n m! } = S( n), so S ( n ) = [,,, S ( n ) ] 6 Then we have S6 n n =,,, S n n = [,,, S( n) S( n) ] S6( n) S6( n) =,,, S6( n),,,, S6( n) =,,, S6( n) S6( n) S ( n ) =,,, m f an only f an [ ] [ ] [ ] () The relatons (7) result from the fact that 7 [ ] n [ m!, ( m+ )! ] Now we may exten the Smaranache functon to the ratonal numbers Every ostve ratonal number a ossesses a unque rme ecomoston of the form a = (6) wth nteger exonents, of whch only a fnte number are nonzero Multlcaton of ratonal numbers s reuce to aton of ther nteger exonent system As a consequence of ths reucton questons concernng vsblty of ratonal numbers are reuce to questons concernng orerng of the corresonng exonent system That s f b = β then b ves a f an only f β for all The greatest common vsor an the least common multle e are gven by = ( a,b, )= mn(,β,), e = [ a,b, ]= max(,β,) (7)

11 Furthermore, the least common multle of nonzero numbers (multlcatvely boune above) s reuce by the rule [ a,b, ]= a, (8) b, to the greatest common vsor of ther recrocal (multlcatvely boune below) Of course we may wrte every ostve ratonal a uner the form a = n / n, wth n an n ostve ntegers * * Defnton 45 The extenson S :Q+ Q+ of the Smaranache functon s efne by S n n = S (n) (9) S 4 (n ) A consequence of ths efnton s that f n an n are ostve ntegers then S n n = S n S n (3) Inee S = S = = = = S S n n n n S 4( ) S n n 4( n) S4( n) S4( n) S4( n) n n an we can mmeately euce that n m S = ( S( n) S( m) ) S S n m n m (3) It results that functon S efne by S( a) = satsfes S a an S n n = S( n ) S( n ) S = S S (3) n n n n for every ostve ntegers n an n Moreover, t results that n n S = ( S( n) S( n) ) S S m m m m an of course the restrcton of S to the ostve ntegers s S 4 The extenson of S to all the ratonales s gven by S( a) = S(a)

12 5 Numercal functons nsre from the efnton of the Smaranache functon We shall use now the equalty () an the relaton (8) to conser numercal functons as the Smaranache functon We may say that m! s the rouct of all ostve smaller than m n the lattce L Analogously the rouct m of all the vsors of m s the rouct of all the elements smaller than m n the lattce L So we may conser functons of the form Θ ( n) = { m/ n ( m) } (33) x It s nown that f m = x x t t then the rouct of all the vsors of m s (m) = m τ (m) where τ(m) = (x + )(x + )(x t + ) s the number of all the vsors of m If n s gven as n () then n (m) f an only f g = x (x + )(x + )(x t + ) g = x (x + )(x + )(x t + ) (34) g t = x t (x + )(x + )(x t + ) t so Θ(n) may be obtane solvng the roblem of non lnear rogrammng x (mn) f = x x t t (35) uner the restrctons (37) The soluton of ths roblem may be obtane alyng the algorthm SUMT (Sequental Unconstrane Mnmzaton Technques) ue to Facco an Mc Cormc [7] Examles For n = 3 4 5, (37) an (38) become (mn) f (x) = 3 x 5x wth x (x + )(x + ) 8, x (x + )(x + ) 4 Conserng the functon U(x,n) = f (x) r = ln g (x), an the system σu / σ x =, σu / σ x = (36) n [7] t s shown that f the soluton x (r), x (r) cannot be exlane from the system we can mae r Then the system becomes x (x + )(x + ) = 8, x (x + )(x + ) = 4 wth the (real) soluton x =, x = 3 mn m/ ρ( m) m = So we have { } Inee ρ(m ) = m τ (m )/ = m 4 = = n For n = 3 567, from the system (39) t results for x the equaton x 3 + 9x + 7x 98 =, wth the real soluton x (,3) It results x ( 4/6, 5/7) s not an amssble Conserng x =, we observe that for x = the ar x, x 7 4 Θ 3 5 = 3 5 soluton of the roblem, but x = 3 gves

13 3 Generally, for n =, from the system (39) t results the equaton x 3 + ( + ) x + x = wth solutons gven by Cartan s formula Of course, usng the metho of the trlets, as for the Smaranache functon, many other functons may be assocate to Θ For the functon ν gven by (8) t s also ossble to generate a class of functon by means of such trlets In the sequel we ll focus the attenton on the analogous of the Smaranache functon an on ts ual n ths case Prooston 5 If n has the ecomoston nto rmes gven by () then () ν(n) = max =,t () ν n n = ν(n ) ν(n ) Proof () Let max = u u Then u u for all,t, so,,, u u j But ( j ) u u, = for j an then n,,, u u Now f for some m < u u we have n [,,, m] [,,, m] () If n =, n = β then, t results the contracton max( β ) = so n n (,, R ) max( β) β ν n n = max = max max, max The functon ν = ν s efne by means of the trlet [ ], where R [ ] = { m/ n [,,, m ]} Its ual, n the sense of the above secton, s the functon efne by the trlet (,, L [ ] ) Let us note ν 4 ths functon ν ( 4 n ) = m [,,, m] n { } That s ν 4 (n) s the greatest natural number wth the roerty that all m ν 4 (n) ve n Let us observe that a necessary an suffcent conton to have ν 4 (n) > s to exst m > such that every rme m ves n From the efnton of ν 4 t also results that ν 4 (n) = m f an only f n s vsble by every n an not by m + Prooston 5 The functon ν 4 satsfes ν 4 n n = ν 4(n ) ν 4 (n ) 3

14 Proof Let us note n = n n,ν 4 (n) = m, ν 4 (n ) = m for =, If m = m m then we rove that m = m From the efnton of ν 4 t results ν ( n ) = m [ m n s vsble by but not by m+ ] 4 If m < m then m + m m so m + ves n an n That s m + ves n If m > m then m + n, so m + ves n But n ves n, so m + ves n If t = max { j n ves n} then ν 4 (n) may be obtane solvng the nteger rogrammng roblem t ( max)f = x ln = x for =,t (37) t ln ln t + x = If f s the maxmal value of f for above roblem, then ν 4 (n) = e f For nstance ν 4 ( ) = 6 Of course, the functon ν may be extene to the ratonal numbers n the same way as Smaranache functon REFERENCES [] M Anre, I Bălăcenou, C Dumtrescu, E Răescu, V Şeleacu A Lnear Combnaton wth the Smaranache Functon to otan the Ientty Proceengs of The 6 th Annual Iranan Mathematcs Conference, (995) [] M Anre, C Dumtrescu, V Seleacu, L Tuţescu, Şt Zamfr Some Remars on the Smaranache Functon Smaranache Functon J 4-5 (994), -5 [3] M Anre, C Dumtrescu, V Seleacu, L Tuţescu, Şt Zamfr La functon e Smaranache, une nouvelle functon ans la theore es nombres - Congress Internatonal H Poncaré, 4-8 May 994, Nancy, France [4] C Dumtrescu Trellus sur es ensembles flous Alcatons a es esaces toologques flous Rev Roum Math Pures Al 3, 986, [5] C Dumtrescu Trells uals Alcatons aux ensembles flous Math Rev Anal Numer Et Theor e l Arox, 5, 986, -6 [6] C Dumtrescu Dual Structures n the Fuzzy Sets Theory an n the Grous Theory Itnerant Sem on Functonal Equatons Arox an Convexty, Cluj- Naoca, Romana, 989, 3-4 [7] Facco an Mc Cormc Nonlnear Programmng Sequental unconstrane Mnmzaton Technque - New Yor, J Wley, 968 [8] P Gronas The Soluton of the Dohantne equaton ση(n) = n, Smaranache Functon J, V 4-5, No (994), 4-6 4

15 [9] H Hasse Number Theory Aaeme-Verlag, Berln, 979 [] L Lovasz Combnatoral Problems an Exercses Aa Kao, Buaest, 979 [] P Raovc-Marculescu Probleme e teora elementară a numerelor E Tehncă, Bucharest, 986 [] E Răescu, N Răescu, C Dumtrescu On the Sumatory Functon assocate to Smaranache Functon, Smaranache Functon J, V 4-5 (994), 7- [3] F Smaranache A functon n the Number Theory An Unv Tmşoara, Ser Şt Math 8 (98),