V. Seleacu

C. Dumitrescu

The Smarandache Function

Erhus University Press

1996

C. DUMITRESCU V. SELEACU

THE SMARANDACHE FUNCTION

ERHUS UNIVERSITY PRESS

1996

"The Smarandache Function", by Constantin Dumitrescu & Vasile Seleacu

Copyright 1996 by C. Dumitrescu & V. Seleacu (Romania) and Erhus University Press 13333 Colossal Cave Road, vail, Box 722, AZ 85641, USA.

(Comments on our books are welcome!

ISBN 1-879585-47-2 Standard Address Number 297-5092

Printed in the United States: of America

Contents

1 The Smarandache Function 5 1.1 Generalised Numerical Scale .........2... 5 1.2 A New Function in Number Theory........ 1] 1.3 Some Formulae for the Calculus of S(n) ..... 18 1.4 Connections with Some Classical Numencal Func- GODS So a es esl ot “Os en ok eer ine a e T A A 27 1.5 The Smarandache Function as Generating Function 36 1.6 Numenical Series Containing the FunctionS ... 48 1.7 Diophantine Equations Involving the Smarandache Function ose an a oS a we es a A 62 1.8 Solved and Unsolved Problems. .......... 73 2 Generalisations of Smarandache Function TT 2.1 Extension to the Set Q of Rational Nnumbers .. 77 2.2 Numerical Functions Inspired from the Definition of Smarandache Function... ........... 86

2.3 Smarandache Functions of First, Second and Third Be be des oS Wee AAS os a BG ee, 1 a 91 2.4 Connections with Fibonacci Sequence. ...... 109 2.5 Solved and Unsolved Problems........... 124

Introduction 3

Introduction

The function named in the title of this book is onginated from the exiled Romanian mathematician Florentin Smarandache, who has significant contributions not only in mathematics, but also in literature. He is the father of The Paradozrist Literary Movement and is the author of many stones, novels, dramas, poems.

The Smarandache function,say S, is a numencal function de- fined such that for every positive integer n, its image S(n) is the smallest positive integer whose factorial is divimble by n.

The results already obtained on this function contain some surprises. Such a surprise is the fact that to expresse Sip) tne exponent «œ is written in a (generalised) numerical scale, say [p], and is read” in another (usual) scale, say (p) (eq. 1.21). More details on this subject may be found im section 1.2.

Another surprise is that "the complement until the identity” (equation 1.34) of S{p*) may be expressed in a dual manner

4 The Smarandache Function

with the exponent of the prime p in the expression of n!, given by Legendre’s formula {eq. 1.15 and eq. 1.36).

Finally, we mention that the Smarandache function may be generalised in various ways, one of these generalisations, the Smarandache function attached to a strong divmibility sequence {eq. 2.59}, and particulary to Fibonacci sequence, has a dual property with the strong divisibility {theorem 2.4.7).

Of course, all these pleasant surprises ate “attractors” for us, the mathematicians, that we are in a permanent search for new wonderful results.

But “the attraction” itself on the mitial concept, started by Florentin Smarandache, permitted to obtain the interesting re- sults mentioned above. Indeed, many mathematicians are al- ready inquired about this subject and have obtained these and other results, permitting the publication of the present book. Among these we mention here Ch. Asbacher, I. Balacenoiu, P. Erdos, H. Ibstedt, P. Gronas, T. Tomita.

We mentione also two of the most interesting problems, still unsolved:

i) Find a formula expressing S{n) by means of n itself and not using the decomposition of the number into primes.

2) Solve the equation S(n) = S(n +1).

The (positive) answer to first of these problems will permet to have more important information on the distnbution of the prime numbers.

Let the future permit to reach the knowledge until these, and other, exciting results.

THE AUTHORS

Chapter 1

The Smarandache Function

1.1 Generalised Numerical Scale

It is said that every positive integer r, strictly greater than 1, determine a numenical scale. That is, given r, every positive integer n may be written under the form:

n = Cmr” + may tir + co (1.1) where m and c; are non-negative integers and 0 < œ < r—1, cm Æ 0. We can attach a symbol to each number from the sequence 0,1,2,...,7—1. These are the digits of the scale, and the equality (1.1) may be written as:

Nr) =. fmm- NH (1.2)

where y 1s the digit symbolsing the number c;. In this manner every integer may be uniquely written in a numerical scale (r) and if we note a; = r', one observe that the

6 The Smarandache Function sequence (a;),en satisfies the recurrence relation

Git = 7A (1.3)

and (1.1) becomes

Nn = CmOm + Cm—14m-1 +... +6141 + code (1.4)

The equality (1.4) may be generalised in the following way. Let (6;)ien be an arbitrary increasing sequence. Then the non- negative integer n may be uniquely wntten under the form:

n = chba + Ch—1bh—1 +... + cibi + Cobo (1.5)

But the conditions satisfied by the digits in this case are not so simple as those from (1.3), satisfied for the scale determined by the sequence (a;)en .

For instance Fibonacci sequence, determined by the condi- tions:

R= =l, Fa = Fait & (1.6)

may be considered as a generalised numerical scale, in the sense

described above. From the inequality

2 F: > Fiya

it results the advantage that the corresponding digits are only O and 1, as for the standard scale determined by r = 2.

So, using the generalised scale determined by Fibonacci se- quence for representing the numbers in the memory of computers we may utilise only two states of the circuits (as when the scale (2) is used) but we need a few memory working with Fibonacci scale, because the digits are less in this case.

Generalised Numerical Scale 7

Another generalised scale, which we shall use in the following, is the scale determined by the sequence

:(p) = ——— 1.7 a;(p) = (1.7) where p > 1 is a prime number. Let us denote this scale by [p]. So we have:

[p]: 1, aa{p), as(p), ...-, ai(p), --- (1.8)

and the corresponding recurrence relation is:

Gi+1(p) = pa;(p) +1 (1.9) This is a relatively simple recurrence, but it is different from

the classical recurrence relation (1.3). Of course, every positive integer may be written as:

Nip] = CmOm(P) + Cm—18m—1(p) + ... + ciai(p) (1.10)

so it may be written in the scale [p].

To determine the conditions satisfied by the digits c; in this case we prove the following lemme:

1.1.1 Lemme. Let n be an arbitrary positive integer. Then for every integer p > 1 the number n may be written uniquely as:

Nn = tian (P) + taan, (p) +... + tran, (p) (1.11) with > m >... > > 0 and

1<tj;<p—1 forj=1,2,..,J-1, l<t;<p (1.12)

8 The Smarandache Function

Proof. From the recurrence relation satisfied by the sequence (a:(p))nene it results:

a;(p) = 1, alp) =1+p, a3(p) =1+ptp”...

So, because

[a;(p), ai42(p)) N [ai (p), ai+2(p)) = 0 it results {[a:(p), aimi) N N*}

Then for every n it exists uniquely n, > 1 such that n E [an (p), @n,4i(p)) and we have

Nee t€

Ne

n= ag eO where [z] denote the integer part of z. If we note kz a an, (p) it results

n = tdm (p) +r with ry < an (p)

H = 0, from the inequalities

On; (p) sn Qn, +1(P) =o (1.13)

it results 1 < ż; <p. If Æ 0, it exists uniquely ng E€ N* such that

rE [Gna (p), An, +1(P))

Generalised Numerical Scale 9

and because an (p) > riit results ni > nz. Also, because

On, +1(P) i hay | <p Gn, (p)

from (1.13) it results 1 < < p—1. Now, it exists uniquely nz such that

tis

Ty = toda, (p) + r2

and so one. After a finite number of steps we obtain:

ri- = klan (p) +r; with r,=0

and < 4-1, 1 < t < p, so the lemme is proved.

Let us observe that in (1.11) unlike from (1.10) all the digits t; are greater than zero. Consequently all the digits c; from (1.10) are between zero and p 1, except the last non-nul digit, which can take also the value p.

If we note by (p) the standard scale determined by the prime number p:

IPP eap (1.14)

it results that the difference between the recurrence relations (1.3) and (1.9) induces essential differences for the calculus in the two scales (p) and [p].

Indeed, as it is proved in [1] if

mis] = 442, nis] = 412 and ris] = 44

then writing

10 The Smarandache Function

mtn+r= 442+ 412 —44_ dcba to determine the digits a,b,c,dwe start the addition from the second column (the column corresponding to a2(5)). We have

Now, using a unit from the first column it results:

5a,(5) + 403(5) = 03(5) + 4a (5)

so (for the moment) b = 4. Continuing, we get:

4a3(5) + 4a3(5) + a3(5) = 5a3(5) + 4a3(5)

and using a new unit from the first column it results:

4a3(5) + 4a3 (5) + a3(5) = a4 (5) + 4a3(5)

soc=4andd=1. Finally, adding the remainder digits:

4a1,(5) + 2a,(5) = 501 (5) + a, (5) = 5a, (5) + 1 = a2 (5) it results that the value of 5 must be modified, and a = 0. So

mtn+r = 145055)

A New Numerical Function 11

1.2 A New Function in Number Theory

This function is the Smarandache function S : N* N* de- fined by the conditions:

(s1) S(n)! is divisible by n, (32) S{n) is the smallest positive integerwith the property (s,)

Let p > 0 be a prime number. We start by the construction of the function

Sp: N N" such that

(ss) Sp(ai(p)) = p' (s4) Hn E N” is written under the form given by (1.11) then Sp(n) = t15,(an: (P)) + t2Sp(ana(p)) +... + tSp(an,(P))

1.2.1 Lemme. For every n N” the exponent of the prime p in the decomposition into primes of n! is greater or equal to n. Proof. From the properties of the integer part we deduce:

ato tte iS] [2 =| Se si 7 Sn) be for every a;,5 N*.

A result does to Legendre assert that the exponent of the prime p in the decomposition into primes of n! is:

n

e(n) = H + B TE (1.15)

Then if n has the decomposition (1.11) it results:

12 The Smarandache Function

[ae tapitti] > [e=] es Bs ges cit [a2] 2 = typ") 4 top} + + tpu}

eee ee eee eee ere ere ere eee eee eee rer ee ee error re er eee eee ery

fanaa > [oga + [a] + + [a = a typu—™ + top™—™ +... + trp?

POCO OREO EOE ROE E RHE ERES ESE SEDO UEEE EEE SEH CEH ETO TH OREEO OTOH ESE DH OL EREHOE

| eee eee | ae = tip? + [ae] +... + [see

and so

[e] + [3] +...+ [2] > tate + ph +... +p) +... +ti{p™ —1 +p™ -2 4.. +p? = = ilan (p) + t2am (p) ++ tian (p) =

1.2.2 Theorem. The function S, defined by the conditions (s3) and (s4) from above satisfies: (1) S,{n)! is divisible by p” (2) S (n) is the smallest positive integer with the property (1).

Proof. The property (1) results from the preceding lemme. To prove (2) let n N* and p > 2 an arbitrary prime.Considering n written as in (1.11) we note

z= typ™ + top™ + ..tp™

A New Numerical Function 13

and we shall prove that the number z is the smallest positive © integer with the property (1).

Indeed, if there exists u N”, u < z such that u! is divisible by p”, then

u <z => u<z-1= (z-1)! is divisible by p”

But

z—1=tp™ + fap" +... +tp™"— l

and n >ng >... >n > 1. Because [k + a] = k + [a] for every integer k, it results:

—] É p | =typ™—* + top) ++ typ th 1

Analogously we have for instance

—1 É | = typ + top + tty pt! WM + tip? -1

[a] = hpm Tl ipm tet pt -i Tmt + bpa] apt ap nn

because 0 < tp™ -l<p-p™—1< p™ +}, Also,

[pete] = fp) + tap? + [aerial = =tp™T™ -1 + typ

The last equality of this kind is:

14 The Smarandache Function

a]

zi top™ +... + typ™ 1 f | =p? [ete = =tp

p™ p™ because

0 < tap™ +-+ tip™ < (p— 1)p™ +... + (p— 1)p™ -1+ p-p™—1<(p—-1) D pit pett—1<(p-1)emt=

t=} Pat = pstl_j <p" —1< p™

Indeed, for the next power of p we have

0

f = 7 = ae + top™ ak, + tip™ a

p™ +1 p™ +1

because O<tip™ +top* +... Htp —1<p“ t 1< pt

From these equahties it results that the exponent of p in the dcomposition into primes of (z 1)! is

[252] + [st] +... + [252] = alph 2 + pu 2 +... +p") +... +tj1(p™ -171 +... + p?) + t¢(p™ t + +p)—np=n-—n<n

and the theorem is proved. Now we may construct the function S : N* N* having the properties (sı) and (s2) as follows:

(t) S(1)=1 (it) For every n= - p5?...p%, with a; > 1, and p; primes, p; # p; we define:

S(n) = max Sp, (æi) (1.16)

A New Numerical Function 15

1.2.3 Theorem. The function S defined by the conditions {t} and (ii) from above satisfies the properties (31) and (32).

Proof. Let us suppose n # 1. We shall note by M(z) an arbitrary multiple of z and

Spy (aig) = max Sp, (a) (1.17) Of course, Spa (2a)! = M(t and because S,;(a;)! = M(p%‘) for i = I, s, it results:

Spa (cig)! = M (pf?) for i = 1,8

Moreover, because p; A p = 1 it results:

Spy (ig)! = M (pi p3p") and so (31) is proved. To prove (s2) let us observe that for every u < Sp, (ai,) we have u! # M(p;,*), because Sp, (ia) is the smallest positive integer with the property k! = M(p;,*). So,

ul # M( pi -pi --p3') = M(n) and the property (s2) is proved. 1.2.4 Proposition. For every prime p the function S, is increasing and surjective, but not injective. The function 5 is generaly increasing, in the sense thai:

(V) ne N (A)REN* S(k) >n

and it is surjective but not injective. 1.2.5 Consequences. 1) For every a N” holds:

S,(a) = S°) (1.18)

16 The Smarandache Function

2) For every n > 4 we have:

nis apime 4> S(n)=n

Indeed, if n > 5 is a prime then S(n) = S,(1) = n.

Conversely, if n > 4 is not a prime but S(n) =n, let n = pi + p?...po with s > 2, a; N*, for i =1,s. Then if Sp, (a) is given by (1.17), from Legendre’s formula (1.15) it results the contradiction:

Spi (Qin) < WigPig CN Also, if n = , with a > 2, it results: S(n) = S,(a)< p-a< p =n

and the theorem is proved. 1.2.6 Examples. 1) If n = 2% . 377 . 733 we have:

S(n) = max{S3(31), S3(27), S7(13)} (1.19)

and to calculate 53(31) we consider the generalised numerical scale

[2]: 1, 3, 7, 31, 63,...

Then 31 = 1- as(2), so S2(31) = 1- 25 = 32. For the calculus of $;(27) we consider the scale

[3]: 1, 4, 13, 40,....

and we have 27 = 2- 13+ 1 = 2a3(3) + 2:(3) so

53(27) = S3(2a3(3) + ai(3)) = 253(a3(3)) + 53(a1 (3)) = =2-334+1-3'=57

Finally, to calculate S7(13) we consider the generalised scale

A New Numerical Function 17

[7]: 1, 8, 57, .. and it results

13 = a,(7) + 5a,(7), 80 S7(13) = 1 - S$,(8) +5-S7(1) = —1.774+5-7= 84

From (1.19) one deduces S({n) = 84. So 84 is the smallest positive integer whose factorial is divisible by 2°! - 377 . 738.

2) Which are the numbers with the factorial ending in 1000 zeros?

To answer this question we observe that for n = 101°% it re- sults S(n)! = M (101%) and S(n) is the smallest positive integer whose factorial endsin 1000 zeros.

We have S(n) = $(210°.51000) = max{53(1000), S;(1000)} = Ss{1000).

Considering the generahsed numerical scale

[5]: 1, 6, 31, 156, 781,... it results:

Ss(1000) = Ss(as(5) + a4(5) + 2a3 (5) + ay (5)) = = 55 + 5t +2. 53+5 = 4005

The numbers 4006. 4007, 4008, 4009 have also the required property, but the factonal of 4010 ends in 1001 zeros.

To calculate S({p*) we need to writte the exponent œ in the generalised scale [p]. For this we observe that:

Om{p) < a => (p™—-1)/(p— 1) < a> p™ < (p— ije +1 4 m < log, ((p Iho + 3)

18 The Smarandache Function

and if

Ap} = kyde(p) + ky—1đy—1 (p) +... + kıa (p) = hy Ky—1..-Ky (1.20)

is the expression of the exponent a in the scale [p], then v is the integer part of log,((p 1)a + 1) and the digit k, is obtained by the equahty

E

Using the same procedure for r,_; it results the next non-zero digit from (1.20)

1.3 Some Formulae for the Calculus of S(n)

From the property (34) satisfied by the function 5S,, one deduce:

S(p*) = plop) (1.21)

that is the value of S(p*) is obtained multiplying the prime p by the number obtained writing the exponent a in the generalised scale [p] and ”reading” it in the usual scale (p).

1.3.1 Example. To calculate $(11!™°) we consider first the generalised scale

{11]: 1, 12, 133, 1464,...

Using the considerations from the end of the preceding sec- tion we get:

1000 = 7a3(11) + 5a2(11) + 9a,(11) = 759ny

Some Formulae for Calculus 19

so $(112) = 11(759) 1) = 11(7-11?+5- 11 +9) = 10021. Con- sequently 10021 is the smallest positive integer whose factorial is divisible by 111°.

The equality (1.21) prove the importance of the scales (p) and [p] for the calculus of S(n).

Let now

u . v v p-l %) =D GPs =D, hair) =P ks U2) = gal j=l

be the expression of the the exponent a in the two scales. It results:

(p la =) k;p > k; j=l y=]

Then noting Op) (a) =); Ci; Spile) =)> k; (1.23) sx=0 j=l

v ; o—1 :

and taking into acount that > kj = p }> kjp ìs exactly j=1 j=0

Plae) one obtain

S(p*) = (9 iera (1.24) * Using the first equality from (1.23) we get:

u

pag) =P) alp? 1)+ Dic s=0

3=0 or

p = J aa =} cia; (p) + ogha)

1=0 p=

20 The Smarandache Function

consequently

—ł ] o = —— (ee) i 4 zaga) (1.25)

where (œp) ip] denote the number obtained writing the exponent

& in the scale (p) and reading it in the scale [p]. . Replacing this expression of œ in (1.24) we get:

sie) = FF (ag) + ala) tola) (126)

One may obtain also a connection between S(p°) and the exponent e,(a) defined by Legendre’s formula (1.15). It is said that e (œ) may be expressed also as:

epla) = a (1.27) so using (1.25) one get:

epla) = (apy) fp] e (1.28) An other formula for ep(œ) may be obtained as follows: if œ given by the first equality from (1.22) is:

Op) = Cap” + Cup" +... + Cp + co (1.29) then because

epla) = [a] + [a] +... + [E] = (cup? + capt? +... + ci) +(cup + Cu1) + Cu

we get:

epla) = ((a co)cpy py = (( H Jo) (1.30)

Some Formulae for Calculus 21

where a») = GiCy—1-.-co 18 the expression of œ in the scale (p). From (1.26) and (1.28) it results:

(pt) = P= (ea) +a) + Foyle) topla) (131)

Using the equalities (1.21) and (1.26) one deduce a connection between the following two numbers:

(œp) jip] = the number œ written in the scale (p) and readed in the scale [p]

(2 1)(p) = the number œ written in the scale [p] and readed in the scale(p)

namely:

P (ano) (2 1) (app = Palae) + (P I a¢p)() (1.32)

To obtain other expressions for S(p*) let us observe that from Legendre’s formula (1.15) it results:

: ; ; œ— il S{p*) = pla ip la)) with 0 < la) < TE (1.33) Then using for S(p®%) the notation S,(a) one obtain: 1 . ea +i,(a)=a (1.34) and so, for each function S, there exists a function t,such that the linear combination (1.34), to obtain the identity, holds.

To obtain expressions of i, let us observe that from (1.27) it results:

22 The Smarandache Function

a = (p— tep(a) + a¢(0) and from (1.24) it results a = (5,(a~) of) (a))/(p 1), 80

(p 1)ep(a) + ogla) = ‘lala alah

S(p*) = (p 1)°e,(a) + (p I)o la) + ogla) (1.35)

Let us return now to the function 1, and observe that from (1.24) and (1.34) it results:

pla) = & = Iple) (1.36) P consequently we can say that there exists a duality between the expression of e (æ) in (1.27) and the above expression of t,(a). One may obtain other connections between 2, and e,.For in- stance from (1.27) and (1.36) it results:

ipla) p (p = 1)e (a) Laai x Tla) (1.37) Also, from

Op = Kykyt..ky = kolp! +p? +... +1) + koa (p? 7+ p+.. tpi)... + kapt ith one obtain

o = (kop’) + koip”™? + ... + kop + ki) + ko (p? 7+ +p°" +... +1) + koa(p? > +p? * +... 1) +...

+k3(p + 1) + ka = (apg) + [2] - [22]

Some Formulae for Calculus 23

that because

[a] = kolp? + peo +... + p + 1) + E + kolp HP H. EPH A + AS to t kalp + 1)+ tatkipi yi and [n + 2] = n + [z]. One obtain

a = (odo) + z| - za] 108)

and we can wnitte:

stor) =p- (|| - (h a9

p and from 1.36) and (1.39) we dededuce

tp(a) = H = on) e (1.40) P P This equality results also directly, from (1.36), taking into acount that stents. [fy P p P P

consequently

P p P

An other expression of i (œ) is obtained from (1.21) and (1.36) or from (1.38) and (1.40). Namely

sf

la) =a— (ati) (1.41)

24 The Smarandache Function

From the definition of the function S it results:

Slol) =p [2] = 0-0,

where œp is the remainder of œ modulo p, and also:

eolSp(a)) >a, ep(Sp(a)—1)<a (143)

Sp(a) a%)(Sp()) TA S la) 1 Tpl Spla) 1) an p-l1 p-1 Using (1.24) it results that S (œ) is the unique solution of the system:

olz) < op (a) < alz- 1) +1 (1.43) At the end of this section we return to the function i,, to - find an asimpthotic behaviour for this “complement until the identity” of the function S,. From the conditions satisfied by this function in (1.33) it results for

Q

Alap) = [2] - ilo)

that A(a,p) > 0. p To find an expression for this function we observe that:

R

and supposing that œ [hp +1, hp+p-— 1] it results (= = [e] so:

Some Formulae for Calculus 25

Aam E 4(a) = |) (145 Also, if œ = Ap, it results

PS ]= [BR ]nact ae [=a

so (1.44) becomes:

A(a,p) = Eg -1 (1.46)

Analogously, if œ = hp + p, one obtains

nn

and [2] = h + 1, so (1.44) has the form (1.46).

It results that for every œ for which A(a,p) has the form (1.45) or (1.46), the value of A(a, p) is maximum if of,);(q) is maximum, so for œ = ay, where

au = (p - 1)(p—1)...e—- 1p Peet ae a

v terms

We have then ia GEO E eee E T

(p 1)(2S + P+ + ES) t+ p= (p° +p?) +... +p? + p) (v 1) = palp) (v 1) It results that ay is not divisible by p if and only if v 1 is not divisible by p. In this case

Talau) = (v-1)(p—1)+p=pv—vt+1

26 The Smarandache Function

and eT A ° ilan) 2 kS i -v that is

3 &u 1 a E 7 V, son f= =n es If v—1€ (hp, hp + p) it results [=] = h, and h(p-1)+1< Afaw,p) < hlp- 1)+p+1

hm A(ay,p) = œ

aun oo

We also observe that

oe aad eo

P P

ofl _ _ ptl tpl

Boned ena a asian p-1 p p—i pal

So, if a4 œ as p” then A(ay, p) œas z.

Also, from

ip(œ) is aop) v

a] afp) - [4]

}

it results

Connections with Classical Functions 27

p(x) =i

P

=]

1.4 Connections with Some Classical. Numerical Functions

In this section we shall present some connections of Smarandache function with Euller’s totient function, von Mangolt’s function, Riemann’s function and the function H(z) denoting the number of primes not greater than z.

1.4.1 Definition. The function of von Mangolt is:

_j lan ifn=p™ A(n) = l gen (1.47)

This function is not a multiplicative function, that is from n Ü m = 1 does not result A(n- m) = A(n)-A(m). For imstance, if n = 3 and m = 5 we have A(n) = In3,A(m) = In5 and A(m-n) = A(15) = 0.

We remember the following results:

1.4.2 Theorem. The following equalities hold:

(2) p A(d)=Inn OA =P wld) in 3

where u is Mobius function, defined by:

1 uni p(n) = 0 if n is divisible by a square (1.48) (-1}* if n py ` o....-Dk

28

The Smarandache Function

1.4.3 Definition. The function Y : R —> R is defined by:

W(z)= 9 bap (1.49)

From the properties of this function we mention only the following two:

1.4.4 The function WV satisfies:

() (2) =F Aln)

(ii) U(r) = In{1, 2, 3,..., [z]

where [1, 2, 3, ..., [z]] denotes the lowest common multiple of 1, 2, 3, ..., [z]

It is said that on the set N* of the positive integers one may consider two latticeal structures:

No=(N",A,V) and Ng=(N"A,V) (1.50)

where

A= min, V = max

A= the greatest common divisor d V= the lowest common multiple

We shall note also n A m = (n,m) and n ¥ m = [n,m].

The order in the lattice N, is noted by < and the order from Na is noted by $ It is said that:

ni L n > n divides n —> nifn (1.51) G

and we also observe that the Smarandache function is not a monotonous function:

< n does not implique S(n,) < S(nz) But, taking into account that

Connections with Classical Functions 29

d S(ny V ng) = S(mi) V S(n2) (1.52) we can consider the function S as a function defined on the lattice Nj with values in the lattice N,: S: Ni N (1.53) In this way the Smarandache function becomes an order pre- serving function, in the sense that:

5 na =>> Shni) < S(n2) (1.54)

It is said [31] that if (V, A, V)is a finite lattice, V = {21, 22,..., zn}, with the induced order <, then for every function f : V R, the corresponding generating function is defined by:

F(n)=)0 Fly) (1.55) yn Now we may return to von Mangolt’s function. Let us observe that to every function:

f:N*—+N* (1.56)

one may attach two generating functions, namely the generating functions F* and determined by the lattices N4 and Mh. Then, by the theorem (1.4.2), for f(z) = A(z) it results:

F4(n) =Y A(k) =Inn (1.57)

kin 4 and also

F°(n) =$ A(k) = ¥(n) = of], 2, .. n]

k<n

30 The Smarandache Function

Then it results the following diagram:

It results a strong connection between the definition of the Smarandache function S and the equalities (1.1) and (2.2) from this diagram.

Let f from (1.56) be the function of von Mangolt’s. Then

Se eee j| = ef (n) = ef) . ef(2). ofl) = etn) n = saa = eF%X)). eF 42) _eF%n)

and so, using the definition of S, we are conducted to consider functions of the form:

Connections with Classical Functions 31

y(n) = min {m / n <[1,2,..m]} (1.58)

We shall study this kind of functions in the section 2.2 of the following chapter.

Returning now to the idea of finding connections between the Smarandache function and some classical numerical functions, we present such a connection, with Euller’s function y. Let us remeber that if p is a prime number then:

y(p*) = p” p% (1.59)

and for œ > 2 we have

p?' =(p—I)oaa(p)+1 so op (p*") =p Using the equality (1.24) it results:

Sp) = (p Ip") + olp) = plp) +p (2.60)

1.4.5 Definition. Let C be the set of all complex num- bers. Then the Dirichlet series attached to a function

f:N* cC D;(2z) p? fn) (1.61)

For some z = z + ty this series may be convergent or not. The simplest Dirichlet series is:

(=o =

asl

32 The Smarandache Function

named the function of Riemann or zeta function. This function converges for Re({z) > 1. It is said that the Diriclet series attached to Möbius function u 18: D(z) = E for Re(z)> 1 ¢(z)

and the Diriclet series attached to Euller’s function ¢ 1s:

D(z) = aaa for Re(z)>2

We also have:

D(z) = (z) for Re(z)>1

where T(n) is the number of divisors of n, including land n. More general,

Dy, (z) = C{z)-((z—k) for Rez) >k+1

where o;(n) is the sum of k“— powers of the divisors of n.

In the sequel we shall writte a(n) instead of o,(n) and r{n) instead of co(n). We also suppose that z = z, so z is a real number.

1.4.6 Theorem. H

is the decomposition of n into primes then the Smarandache function and Riemann’s function are linked by the following equahty:

C= 1) yn ty he -ri ¢(z) ada p:

(1.62)

Connections with Classical Functions 33

Proof. We have seen that between the functions y and ¢ there exists a connection given by:

((z-1)_ > ofr) C{z) Bess nz (1.63)

Moreover,

oln) olp?) =i (Spl?) - pi)

and replacing this expression of p(n) in (1.63) it results the equality (1.62). The Dirichlet series corresponding to the function S is:

and noting by D Fa the Dirichlet series attached to the generating function it results: 1.4.7 Theorem. For every z > 2 we have:

(1) ¢(z) < Ds(z) < ¢(z 1) (tt) C*(z) < Drg(z) < C(z) e(z- 1)

Proof. The inequalities (7) result from the fact that

1 < S(n) <n for every n E€ N” (1.64) (tz) We have:

C(e): Ds(z) =(X $) = 991) + TO + Sts) + SOSO) +..... = Dpa(z)

and the inequalities results using (2). One observe that (ti) is equivalent with

34 The Smarandache Function

D-(2) < Deg (2) < De(2)

This equality may be also deduced observing that from (1.64) it results:

YW 1<) Sk) <} k k<n k<n k<n

and consequently:

r(n) < F$(n) < a(n) (1.65) In [19] has been proved for even that:

r(n) < F(a) Sn+4

To prove other inequalities satisfied by the Dirichlet series Ds we remember first that if f and g are two unbounded functions defined on the set R of real numbers satisfying g({z) > 0, and if there exist the constants C,, C2 such that

/f(z)/ < Cyg(z) for every z > C3

then the functions f and g are said to be of the same order of magnitude and one note

f(z) = O(g(z)) Particularly, is noted by O(1) any function which is bounded

for z > Ca. The fact that it exists

f(z) Ae g(z)

=0

is noted by

Generating Functions 35

f(z) = o(9(z))

Particularly is noted by o{1) any function tending to zero when z tends to infinity and evidently we have:

f(z) = ofg(z)) => f(z) = A(g(z)) It is said that Rieman’s function satisfies the properties given

bellow: 1.4.8 Theorem. For all complex number z we have:

(2) ¢(z) + Ay + O(1) (ii) In¢(z) = ln +, + O(z - 1)

(ii) C(e) = -rrr +001) Using the theorems (1.4.7) and (1.4.8) now we obtain:

1.4.9 Theorem. The Dirichlet series Ds attached to the Smarandache function S and his derivative D% satisfy:

(i) + Ons < Ds(z) < 45+ Ol) (i) gy + O(1) < D; TA < -gy +0(1)

The number of primes not exceeding a given number z is usually denoted by H(z}. In [39] is given a connection between the Smarandache function S and the function I.

Starting from the fact that S(n) < n for every n and that, for n > 4 we have S(n) = n if and only if n is a prime, it is obtained the equality:

H(z) = >> ; 2] - 1

36 The Smarandache Function

1.5 The Smarandache Function as Generating Function

It is said that Mobius inversion formula permet to obtain any numerical function f from his generating function F?. Namely,

f(n) =D wa) F4(5) (1.66) djn if F4(n) =F f(d) djn

So, we can consider every numerical function f in two distinct positions: one is that in which we are interested to consider its generating function, and in the second we consider the function f itself as a generating function, for some numerical function g.

a(n) = DEG) D Fin) =F fd) (1.67)

djn

For instance if f(n) = n is the identity map of N” we get:

g(n) 2 MdG =el); Fn) =} d=old) (1.68)

djn

In the case when f is the Smarandache function S, it is dif- ficult to calculate for any positive integer n the value of F4(n). That because :

F3(n) =$ S(d) =} max{s(5") (1.69)

djn djn

where 6; are the prime factors of d.

Generating Functions 37

However, there are two situations in which the explicite forme of F¢(n)} may be obtained easily. These are for n = and for n a square free number.

In the first case we have

Fi) Spi) (@ - i + ol) =

a l (1.70) = (p— 1) D4 È opi) I= Let consider n = py - p2.....py a square free number, where < p2 <... < pk are the prime factors of n. It resulta:

S(n) = pk and F#(1) = SQ) =1 F§(p1) = S(1) + S(p) = 14+ pr F§$(p1 - p2) = S(1) + S(pi) + S(p2) + S(pr - p2) = 14+ +2p2 F$(py + pa: ps) = 1+ pr + 2p: + 2 ps + F(p: p2) + 2 ps and also: Fé(n) =l + F4(p, * Pa.-..-- Phi) + 257o

Then

k Fé(n) = 1+ ye (1.71) One observe that because S(n) = ph, replacing the values of F(t) given by (1.71) in

S(n) = 5y u(r) Fé(t) (1.72)

rtan apparently we get an expression of the prime number p; by means of the preceding primes pı, 92,...pp-1. In reality (1.72) is an

38 The Smarandache Function

identity in which ,after the reduction of all similar terms, the prime numbers p; has the coefficient equal to zero. In [19] it is solved the equation Fé(n) =n (1.73) under the hypothesis

SQ) =0 (1.74)

and it is found the following result:

1.5.1 Proposition. The equation (1.73) has as solutions only: all the prime numbers n and the composit numbers n = 9, 16, 24.

Proof. Because

Fi(n) =F S(d) (1.75)

djn

under the hypothesis (1.74) one observe that every prime is a solution of our equation. Let now suppose n > 4 be a composit