SOME SMARANDACHE-TYPE MULTIPLICATIVE FUNCTIONS

Henry Bottomley

5 Leydon Close, London SE16 5PF, United Kingdom

E-mail:  SE16@btinternet.com 

This note considers eleven particular families of interrelated multiplicative functions, many of which are listed in Smarandache's problems.

These are multiplicative in the sense that a function f(n) has the property that for any two coprime positive integers a and b, i.e. with a highest common factor (also known as greatest common divisor) of 1, then f(a*b)=f(a)*f(b). It immediately follows that f(1)=1 unless all other values of f(n) are 0. An example is d(n), the number of divisors of n. This multiplicative property allows such functions to be uniquely defined on the positive integers by describing the values for positive integer powers of primes. d(pi)=i+1 if i>0; so d(60) = d(22*31*51) = (2+1)*(1+1)*(1+1) = 12.

Unlike d(n), the sequences described below are a restricted subset of all multiplicative functions. In all of the cases considered here, f(pi)=pg(i) for some function g which does not depend on p.

 

Definition

Multiplicative with p^i->p^...

Am(n)

Number of solutions to xm == 0 (mod n)

i-ceiling[i/m]

Bm(n)

Largest mth power dividing n

m*floor[i/m]

Cm(n)

mth root of largest mth power dividing n

floor[i/m]

Dm(n)

mth power-free part of n

i-m*floor[i/m]

Em(n)

Smallest number x (x>0) such that n*x is a perfect mth power (Smarandache mth power complements)

m*ceiling[i/m]-i

Fm(n)

Smallest mth power divisible by n divided by largest mth power which divides n

m*(ceiling[i/m]-floor[i/m])

Gm(n)

mth root of smallest mth power divisible by n divided by largest mth power which divides n

ceiling[i/m]-floor[i/m]

Hm(n)

Smallest mth power divisible by n (Smarandache ^m function (numbers))

m*ceiling[i/m]

Jm(n)

mth root of smallest mth power divisible by n (Smarandache Ceil Function of mth Order)

ceiling[i/m]

Km(n)

Largest mth power-free number dividing n (Smarandache mth power residues)

min(i,m-1)

Lm(n)

n divided by largest mth power-free number dividing n

max(0,i-m+1)

Relationships between the functions

Some of these definitions may appear to be similar; for example Dm(n) and Km(n), but for m>2 all of the functions are distinct (for some values of n given m). The following relationships follow immediately from the definitions:

 

Bm(n)=Cm(n)m

(1)

 

n=Bm(n)*Dm(n)

(2)

 

Fm(n)=Dm(n)*Em(n)

(3)

 

Fm(n)=Gm(n)m

(4)

 

Hm(n)=n*Em(n)

(5)

 

Hm(n)=Bm(n)*Fm(n)

(6)

 

Hm(n)=Jm(n)m

(7)

 

n=Km(n)*Lm(n)

(8)

These can also be combined to form new relationships; for example from (1), (4) and (7) we have

 

Jm(n)=Cm(n)*Gm(n)

(9)

Further relationships can also be derived easily. For example, looking at Am(n), a number x has the property xm==0 (mod n) if and only if xm is divisible by n or in other words a multiple of Hm(n), i.e. x is a multiple of Jm(n). But Jm(n) divides into n, so we have the pretty result that:

 

n=Jm(n)*Am(n)

(10)

We could go on to construct a wide variety of further relationships, such as using (5), (7) and (10) to produce:

 

nm-1=Em(n)*Am(n)m

(11)

but instead we will note that Cm(n) and Jm(n) are sufficient to produce all of the functions from Am(n) through to Jm(n):

 

Am(n)=n/Jm(n)

(12)

 

Bm(n)=Cm(n)m

 

 

Cm(n)=Cm(n)

 

 

Dm(n)=n/Cm(n)m

(13)

 

Em(n)=Jm(n)m/n

(14)

 

Fm(n)=(Jm(n)/Cm(n))m

(15)

 

Gm(n)=Jm(n)/Cm(n)

(16)

 

Hm(n)=Jm(n)m

 

 

Jm(n)=Jm(n)

 

Clearly we could have done something similar by choosing one element each from two of the sets {A,E,H,J}, {B,C,D}, and {F,G}. The choice of C and J is partly based on the following attractive property which further deepens the multiplicative nature of these functions.

 

If m=a*b then:

 

 

Cm(n)=Ca(Cb(n))

(17)

 

Jm(n)=Ja(Jb(n))

(18)

Duplicate functions when m=2 ...

When m=2, D2(n) is square-free and F2(n) is the smallest square which is a multiple of D2(n), so

 

F2(n)=D2(n)2

(19)

Using (3) and (4) we then have:

 

D2(n)=E2(n)=G2(n)

(20)

and from (13) and (14) we have

 

n=C2(n)*J2(n)

(21)

so from (10) we get

 

A2(n)=C2(n)

(22)

... and when m=1

If m=1, all the functions described either produce 1 or n. The analogue of (20) is still true with

 

D1(n)=E1(n)=G1(n)=1

(23)

but curiously the analogue of (22) is not, since:

 

A1(n)=1

(24)

 

C1(n)=n

(25)

The two remaining functions

All this leaves two slightly different functions to be considered: Km(n) and Lm(n). They have little connection with the other sequences except for the fact that since Gm(n) is square-free, and divides Dm(n), Em(n), Fm(n), and Gm(n), none of which have any factor which is a higher power than m, we get:

 

Gm(n)=Jm(Dm(n))=Jm(Em(n))=Jm(Fm(n))=Jm(Gm(n))=K2(Dm(n))=K2(Em(n))=K2(Fm(n))=K2(Gm(n))

(26)

and so with (8) and (10)

 

n/Gm(n)=Am(Dm(n))=Am(Em(n))=Am(Fm(n))=Am(Gm(n))=L2(Dm(n))=L2(Em(n))=L2(Fm(n))=L2(Gm(n))

(27)

We also have the related convergence property that for any y, there is a z (e.g. floor[log2(n)]) for which

 

Gm(n)=Jm(n)=K2(n) for any n<=y and any m>z

(28)

 

Am(n)=L2(n) for any n<=y and any m>z

(29)

There is a simple relation where

 

Lm(n)=La(Lb(n)) if m+1=a+b and a,b>0

(29)

and in particular

 

Lm(n)=Lm-1(L2(n)) if m>1

(30)

 

L3(n)=L2(L2(n))

(31)

 

L4(n)=L2(L2(L2(n)))

(32)

so with (8) we also have

 

Km(n)=Kb(n)*Ka(n/Kb(n)) if m+1=a+b and a,b>0

(33)

 

Km(n)=Km-1(n)*K2(n/Km-1(n)) if m>1

(34)

 

K3(n)=K2(n)*K2(n/K2(n))

(35)

 

K4(n)=K2(n)*K2(n/K2(n))*K2(n/(K2(n)*K2(n/K2(n))))

(36)

Recording the functions

The values of all these functions for n up from n=1 to about 70 or more are listed in Neil Sloane's Online Encylopedia of Integer Sequences for m=2, 3 and 4:

 

m=1

m=2

m=3

m=4

m>=x and n<2x

Am(n)

1

A000188

A000189

A000190

L2(n)

Bm(n)

n

A008833

A008834

A008835

1

Cm(n)

n

A000188

A053150

A053164

1

Dm(n)

1

A007913

A050985

A053165

n

Em(n)

1

A007913

A048798

A056555

K2(n)m/n

Fm(n)

1

A055491

A056551

A056553

K2(n)m

Gm(n)

1

A007913

A056552

A056554

K2(n)

Hm(n)

n

A053143

A053149

A053167

K2(n)m

Jm(n)

n

A019554

A019555

A053166

K2(n)

Km(n)

1

A007947

A007948

A058035

n

Lm(n)

n

A003557

A062378

A062379

1

Further reading

K. Atanassov, On the 22-nd, the 23-th, and the 24-th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 80-82.

K. Atanassov, On Some of the Smarandache's Problems, American Research Press, 1999, 16-21.

M. L. Perez et al., eds., Smarandache Notions Journal

M. Popescu, M. Nicolescu, About the Smarandache Complementary Cubic Function, Smarandache Notions Journal, Vol. 7, No. 1-2-3, 1996, 54-62.

F. Russo An Introduction to the Smarandache Square Complementary Function, American Research Press

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, 2001 http://www.research.att.com/~njas/sequences/

F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.

F. Smarandache, "Collected Papers", Vol. II, Tempus Publ. Hse, Bucharest, 1996.

H. Ibstedt Surfing on the Ocean of Numbers, American Research Press, 27-30

E. W. Weisstein, MathWorld, 2000 http://mathworld.wolfram.com/ Cubic Part, Squarefree Part, Cubefree Part, Smarandache Ceil Function

Multiplicative is not used here in the same sense as in S Tabirca, About Smarandache-Multiplicative Functions, American Research Press.