Type: Article

Arithmetic functions monotonic at consecutive arguments

De Koninck, JM; Luca, Florian;

Abstract:

For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer k >= 2, the string of inequalities f(n +1) < f(n + 2) < ... < f(n + k) holds for infinitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for k = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying sigma(2) (n - 1) < sigma(2) (n) < sigma(2) (n + 1), where sigma 2(n) = Sigma(d vertical bar n)d2. On the other hand, we prove that for the function f(n) := Sigma(p vertical bar n)p2 we do have f(n - 1) < f(n) < f(n + 1) infinitely often.