Residue Classes Free of Values of Euler's Function

2020

Euler’s totient function counts the positive integers up to a given integer n that are relatively prime to n. The aim of this article is to give a result about the sum : n ∑ k=1 p|k φ(k) , for every prime number p .

Proceedings - Mathematical Sciences

In this article, we investigate sparse subsets of the natural numbers and study the sparseness of some sets associated to the Euler's totient function φ via the property of 'Banach Density'. These sets related to the totient function are defined as follows: V := φ(N) and N i := {N i (m) : m ∈ V } for i = 1, 2, 3, where N 1 (m) = max{x ∈ N : φ(x) ≤ m}, N 2 (m) = max(φ −1 (m)) and N 3 (m) = min(φ −1 (m)) for m ∈ V. Masser and Shiu call the elements of N 1 as 'sparsely totient numbers' and construct an infinite family of these numbers. Here we construct several infinite families of numbers in N 2 \ N 1 and an infinite family of composite numbers in N 3. We also study (i) the ratio N 2 (m) N 3 (m) , which is linked to the Carmichael's conjecture, namely, |φ −1 (m)| ≥ 2 ∀ m ∈ V , and (ii) arithmetic and geometric progressions in N 2 and N 3. Finally, using the above sets associated to the totient function, we generate an infinite class of subsets of N, each with asymptotic density zero and containing arbitrarily long arithmetic progressions.

The Ramanujan Journal, 2005

arXiv: Number Theory, 2019

Let $N_1(m)=\max\{n \colon \phi(n) \leq m\}$ and $N_1 = \{N_1(m) \colon m \in \phi(\mathbb{N})\}$ where $\phi(n)$ denotes the Euler's totient function. Masser and Shiu \cite{masser} call the elements of $N_1$ as `sparsely totient numbers' and initiated the study of these numbers. In this article, we establish several results for sparsely totient numbers. First, we show that a squarefree integer divides all sufficiently large sparsely totient numbers and a non-squarefree integer divides infinitely many sparsely totient numbers. Next, we construct explicit infinite families of sparsely totient numbers and describe their relationship with the distribution of consecutive primes. We also study the sparseness of $N_1$ and prove that it is multiplicatively piecewise syndetic but not additively piecewise syndetic. Finally, we investigate arithmetic/geometric progressions and other additive and multiplicative patterns like $\{x, y, x+y\}, \{x, y, xy\}, \{x+y, xy\}$ and their generali...

Glasgow Mathematical Journal, 1991

Following Masser and Shiu [6] we say that a positive integer n is sparsely totient ifHere φ is the familiar Euler totient function. We write ℱ for the set of sparsely totient numbers. In [6] several results are proved about the multiplicative structure of ℱ. If we write P(n) for the largest prime factor of n then it was shown (Theorem 2 of [6]) thatand infinitely often

Journal of Integer Sequences

We find the form of all solutions to ϕ(n)|σ(n) with three or fewer prime factors, except when the quotient is 4 and n is even.

Journal of Number Theory, 2009

The expressions φ(n) + σ (n) − 3n and φ(n) + σ (n) − 4n are unusual among linear combinations of arithmetic functions in that they each vanish on a nonempty set of composite numbers. In 1966, Nicol proved that the set A := {n | (φ(n) + σ (n))/n ∈ N 3 } contains 2 a · 3 · (2 a−2 · 7 − 1) if and only if 2 a−2 · 7 − 1 is prime and conjectured that A contains no odd integers. A 2008 paper by Luca and Sandor completely classifies the elements of A that have three distinct prime factors and observes that Nicol's conjecture holds for numbers with fewer than six distinct prime factors. In this paper we let A K denote the set of n ∈ A with exactly K distinct prime factors and present a computer-implementable algorithm that decides whether Nicol's conjecture holds for a given A K . Using this algorithm, we verify Nicol's conjecture for A 6 and completely classify the elements of A 4 . We prove that all but finitely many n ∈ A 4 have the form 2 a · 3 · p 3 · p 4 , and that all but finitely many n ∈ A 5 are divisible by 6 and not 9. In addition, we prove that every A K is contained in a finite union of sequences that each have the form {p

Asian-European Journal of Mathematics

The main goal of this paper is to provide a group theoretical generalization of the well-known Euler’s totient function. This determines an interesting class of finite groups.

arXiv (Cornell University), 2020

We study the set D of positive integers d for which the equation φ(a)−φ(b) = d has infinitely many solution pairs (a, b). We show that min D 154, exhibit a specific A so that every multiple of A is in D, and show that any progression a mod d with 4|a and 4|d, contains infinitely many elements of D. We also show that the Generalized Elliott-Halberstam Conjecture, as defined in [6], implies that D contains all positive, even integers.

Acta Arithmetica, 2021