A Note on Some Characterizations of Arithmetic Functions

Some Characterizations of Functions Computable in On-Line Arithmetic

Absfmct-After a short introduction to on-line computing, we prove that the functions computable in on-line by a finite automaton are piecewise afthe functions whose coefikients are rational numbers (i.e., the fnnctions f(s) = UP + b, or f(z-, y) = nx + by + c where a. b, and c are rational). A consequence of this study is that multiplication, division, and elementary functions of operands of ar...

A Note on Rational Arithmetic Functions

On the Iterates of Some Arithmetic Functions

1. Introduction. For any arithmetic function f(n), we denote its iterates as follows : f l (n) = f (n) , fk (n) = fl[ fk_1(n) ] (k > 1). Let a(n) and a* (n) denote,'respectívely, the sum of the divisors of n, and the sum of its unitary divisors, where we recall that d is called a unitary divisor of n if (d,n/d) = 1. Makowski and Schinzel [3] proved that and conjectured that ak (n) lim inf n < o...

On the Coprimality of Some Arithmetic Functions

Let φ stand for the Euler function. Given a positive integer n, let σ(n) stand for the sum of the positive divisors of n and let τ(n) be the number of divisors of n. We obtain an asymptotic estimate for the counting function of the set {n : gcd(φ(n), τ(n)) = gcd(σ(n), τ(n)) = 1}. Moreover, setting l(n) := gcd(τ(n), τ(n+ 1)), we provide an asymptotic estimate for the size of #{n 6 x : l(n) = 1}.

On characterizations of weakly $e$-irresolute functions

The aim of this paper is to introduce and obtain some characterizations of weakly $e$-irresolute functions by means of $e$-open sets defined by Ekici [6]. Also, we look into further properties relationships between weak $e$-irresoluteness and separation axioms and completely $e$-closed graphs.