SOME
SMARANDACHETYPE MULTIPLICATIVE FUNCTIONS
Email: 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(p^{i})=i+1 if i>0; so d(60) = d(2^{2}*3^{1}*5^{1}) = (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(p^{i})=p^{g(i)} for some function g which does not depend on p.

Definition 
Multiplicative with p^i>p^... 
A_{m}(n) 
Number of solutions to x^{m} == 0 (mod n) 
iceiling[i/m] 
B_{m}(n) 
Largest m^{th} power dividing n 
m*floor[i/m] 
C_{m}(n) 
m^{th} root of largest mth power dividing n 
floor[i/m] 
D_{m}(n) 
m^{th} powerfree part of n 
im*floor[i/m] 
E_{m}(n) 
Smallest number x (x>0) such that n*x is a perfect m^{th} power (Smarandache m^{th} power complements) 
m*ceiling[i/m]i 
F_{m}(n) 
Smallest m^{th} power divisible by n divided by largest m^{th} power which divides n 
m*(ceiling[i/m]floor[i/m]) 
G_{m}(n) 
m^{th} root of smallest m^{th} power divisible by n divided by largest m^{th} power which divides n 
ceiling[i/m]floor[i/m] 
H_{m}(n) 
Smallest m^{th} power divisible by n (Smarandache ^m function (numbers)) 
m*ceiling[i/m] 
J_{m}(n) 
m^{th} root of smallest mth power divisible by n (Smarandache Ceil Function of m^{th} Order) 
ceiling[i/m] 
K_{m}(n) 
Largest m^{th} powerfree number dividing n (Smarandache m^{th} power residues) 
min(i,m1) 
L_{m}(n) 
n divided by largest m^{th} powerfree number dividing n 
max(0,im+1) 
Some of these definitions may appear to be similar; for example D_{m}(n) and K_{m}(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:

B_{m}(n)=C_{m}(n)^{m} 
(1) 

n=B_{m}(n)*D_{m}(n) 
(2) 

F_{m}(n)=D_{m}(n)*E_{m}(n) 
(3) 

F_{m}(n)=G_{m}(n)^{m} 
(4) 

H_{m}(n)=n*E_{m}(n) 
(5) 

H_{m}(n)=B_{m}(n)*F_{m}(n) 
(6) 

H_{m}(n)=J_{m}(n)^{m} 
(7) 

n=K_{m}(n)*L_{m}(n) 
(8) 
These can also be combined to form new relationships; for example from (1), (4) and (7) we have

J_{m}(n)=C_{m}(n)*G_{m}(n) 
(9) 
Further relationships can also be derived easily. For example, looking at A_{m}(n), a number x has the property x^{m}==0 (mod n) if and only if x^{m} is divisible by n or in other words a multiple of H_{m}(n), i.e. x is a multiple of J_{m}(n). But J_{m}(n) divides into n, so we have the pretty result that:

n=J_{m}(n)*A_{m}(n) 
(10) 
We could go on to construct a wide variety of further relationships, such as using (5), (7) and (10) to produce:

n^{m1}=E_{m}(n)*A_{m}(n)^{m} 
(11) 
but instead we will note that C_{m}(n) and J_{m}(n) are sufficient to produce all of the functions from A_{m}(n) through to J_{m}(n):

A_{m}(n)=n/J_{m}(n) 
(12) 

B_{m}(n)=C_{m}(n)^{m} 


C_{m}(n)=C_{m}(n) 


D_{m}(n)=n/C_{m}(n)^{m} 
(13) 

E_{m}(n)=J_{m}(n)^{m}/n 
(14) 

F_{m}(n)=(J_{m}(n)/C_{m}(n))^{m} 
(15) 

G_{m}(n)=J_{m}(n)/C_{m}(n) 
(16) 

H_{m}(n)=J_{m}(n)^{m} 


J_{m}(n)=J_{m}(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: 


C_{m}(n)=C_{a}(C_{b}(n)) 
(17) 

J_{m}(n)=J_{a}(J_{b}(n)) 
(18) 
When m=2, D_{2}(n) is squarefree and F_{2}(n) is the smallest square which is a multiple of D_{2}(n), so

F_{2}(n)=D_{2}(n)^{2} 
(19) 
Using (3) and (4) we then have:

D_{2}(n)=E_{2}(n)=G_{2}(n) 
(20) 
and from (13) and (14) we have

n=C_{2}(n)*J_{2}(n) 
(21) 
so from (10) we get

A_{2}(n)=C_{2}(n) 
(22) 
If m=1, all the functions described either produce 1 or n. The analogue of (20) is still true with

D_{1}(n)=E_{1}(n)=G_{1}(n)=1 
(23) 
but curiously the analogue of (22) is not, since:

A_{1}(n)=1 
(24) 

C_{1}(n)=n 
(25) 
All this leaves two slightly different functions to be considered: K_{m}(n) and L_{m}(n). They have little connection with the other sequences except for the fact that since G_{m}(n) is squarefree, and divides D_{m}(n), E_{m}(n), F_{m}(n), and G_{m}(n), none of which have any factor which is a higher power than m, we get:

G_{m}(n)=J_{m}(D_{m}(n))=J_{m}(E_{m}(n))=J_{m}(F_{m}(n))=J_{m}(G_{m}(n))=K_{2}(D_{m}(n))=K_{2}(E_{m}(n))=K_{2}(F_{m}(n))=K_{2}(G_{m}(n)) 
(26) 
and so with (8) and (10)

n/G_{m}(n)=A_{m}(D_{m}(n))=A_{m}(E_{m}(n))=A_{m}(F_{m}(n))=A_{m}(G_{m}(n))=L_{2}(D_{m}(n))=L_{2}(E_{m}(n))=L_{2}(F_{m}(n))=L_{2}(G_{m}(n)) 
(27) 
We also have the related convergence property that for any y, there is a z (e.g. floor[log_{2}(n)]) for which

G_{m}(n)=J_{m}(n)=K_{2}(n) for any n<=y and any m>z 
(28) 

A_{m}(n)=L_{2}(n) for any n<=y and any m>z 
(29) 
There is a simple relation where

L_{m}(n)=L_{a}(L_{b}(n)) if m+1=a+b and a,b>0 
(29) 
and in particular

L_{m}(n)=L_{m1}(L_{2}(n)) if m>1 
(30) 

L_{3}(n)=L_{2}(L_{2}(n)) 
(31) 

L_{4}(n)=L_{2}(L_{2}(L_{2}(n))) 
(32) 
so with (8) we also have

K_{m}(n)=K_{b}(n)*K_{a}(n/K_{b}(n)) if m+1=a+b and a,b>0 
(33) 

K_{m}(n)=K_{m1}(n)*K_{2}(n/K_{m1}(n)) if m>1 
(34) 

K_{3}(n)=K_{2}(n)*K_{2}(n/K_{2}(n)) 
(35) 

K_{4}(n)=K_{2}(n)*K_{2}(n/K_{2}(n))*K_{2}(n/(K_{2}(n)*K_{2}(n/K_{2}(n)))) 
(36) 
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<2^{x} 
A_{m}(n) 
1 
L_{2}(n) 

B_{m}(n) 
n 
1 

C_{m}(n) 
n 
1 

D_{m}(n) 
1 
n 

E_{m}(n) 
1 
K_{2}(n)^{m}/n 

F_{m}(n) 
1 
K_{2}(n)^{m} 

G_{m}(n) 
1 
K_{2}(n) 

H_{m}(n) 
n 
K_{2}(n)^{m} 

J_{m}(n) 
n 
K_{2}(n) 

K_{m}(n) 
1 
n 

L_{m}(n) 
n 
1 
Further reading
K. Atanassov, On the 22nd, the 23th, and the 24th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 8082.
K. Atanassov, On Some of the Smarandache's Problems, American Research Press, 1999, 1621.
I. Balacenoiu et al., eds., Smarandache Notions Journal
M. Popescu, M. Nicolescu, About the Smarandache Complementary Cubic Function, Smarandache Notions Journal, Vol. 7, No. 123, 1996, 5462.
F. Russo An Introduction to the Smarandache Square Complementary Function, American Research Press
N. J. A. Sloane, The OnLine Encyclopedia of Integer Sequences, 2001 http://www.research.att.com/~njas/sequences/
F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., PhoenixChicago, 1993.
F. Smarandache, "Collected Papers", Vol. II, Tempus Publ. Hse, Bucharest, 1996.
H. Ibstedt Surfing on the Ocean of Numbers, American Research Press, 2730
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 SmarandacheMultiplicative Functions, American Research Press.