Infinite families of noncototients

Abstract

Abstract. For any positive integer n let φ(n) be the Euler function of n. A positive integer n is called a noncototient if the equation x − φ(x) = n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression (2 m k) m≥1 consists entirely of noncototients. We then use computations to detect seven such positive integers k.

On some problems of Mąkowski–Schinzel and Erdős concerning the arithmetical functions φ and σ

“…Recall that an integer n is called a cototient if it is in the range of the function u, that is, n = m − φ(m) for some integer m. It is not known if the set of cototients has an asymptotic density nor if the upper density of the set is < 1. In fact, until a few years ago it was not even known that there are infinitely many non-cototients, until an infinite family of such was pointed out by Browkin and Schinzel in [3] (see also [12] for more examples of such infinite families of non-cototients). In analogy with the notion of a cototient, let us call a positive integer n a strong cototient if the equation u k (x) = n has a positive solution x for every k ≥ 1.…”

When the Sum of Aliquots Divides the Totient

“…and let s(n) = σ(n) − n be the sum of the aliquot divisors of n 1. The function s(n) and related arithmetic functions (such as f (n) = n − ϕ(n)) have been previously studied in the literature (see, for example, [1,2,[5][6][7][8]).…”

On the number of nonquadratic residues which are not primitive roots

“…Throughout, we use p and q for prime numbers. A related question from [3] regarding whether or not there exist infinitely many positive integers m not in the range of the function n − φ(n) has been treated in [1] and [2]. Proof.…”