EUDML

Let φ(·) and σ(·) denote the Euler function and the sum of divisors function, respectively. We give a lower bound for the number of m ≤ x for which the equation m = σ(n) - n has no solution. We also show that the set of positive integers m not of the form (p-1)/2 - φ(p-1) for some prime number p has a positive lower asymptotic density.

Prime numbers with Beatty sequences

William D. Banks; Igor E. Shparlinski — 2009

Colloquium Mathematicae

A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of infinitely many primes which are not of the form p = 2⌊αn⌋ + 1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic...

Carmichael numbers composed of primes from a Beatty sequence

William D. Banks; Aaron M. Yeager — 2011

Colloquium Mathematicae

Let α,β ∈ ℝ be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence $ℬ_{α, β} = {(⌊ α n + β ⌋)}_{n = 1}^{\infty}$ . We conjecture that the same result holds true when α is an irrational number of infinite type.

Average Value of the Euler Function on Binary Palindromes

William D. Banks; Igor E. Shparlinski — 2006

Bulletin of the Polish Academy of Sciences. Mathematics

We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if $ℬ_{2 ℓ}$ denotes the set of binary palindromes with precisely 2ℓ binary digits, we derive an asymptotic formula for the average value of the Euler function on $ℬ_{2 ℓ}$ .

On integers with a special divisibility property

William D. Banks; Florian Luca — 2006

Archivum Mathematicum

In this note, we study those positive integers $n$ which are divisible by $\sum_{d | n} λ (d)$ , where $λ (\cdot)$ is the Carmichael function.

Consecutive primes in tuples

William D. Banks; Tristan Freiberg; Caroline L. Turnage-Butterbaugh — 2015

Acta Arithmetica

In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple $(x) = {g x + h_{j}}_{j = 1}^{k}$ of linear forms in ℤ[x], the set $(n) = {g n + h_{j}}_{j = 1}^{k}$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $(n) = {g n + h_{j}}_{j = 1}^{k}$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps $δ_{1}, . . ., δ_{m}$ form an increasing (resp....

Representations of integers as sums of primes from a Beatty sequence

William D. Banks; Ahmet M. Güloğlu; C. Wesley Nevans — 2007

Acta Arithmetica

On rough and smooth neighbors.

William D. Banks; Florian Luca; Igor E. Shparlinski — 2007

Revista Matemática Complutense

We study the behavior of the arithmetic functions defined by F(n) = P⁺(n) / P^-(n+1) and G(n) = P⁺(n+1) / P^-(n) (n ≥ 1) where P⁺(k) and P^-(k) denote the largest and the smallest prime factors, respectively, of the positive integer k.

Waring’s problem for Beatty sequences and a local to global principle

William D. Banks; Ahmet M. Güloğlu; Robert C. Vaughan — 2014

Journal de Théorie des Nombres de Bordeaux

We investigate in various ways the representation of a large natural number $N$ as a sum of $s$ positive $k$ -th powers of numbers from a fixed Beatty sequence. , a very general form of the local to global principle is established in additive number theory. Although the proof is very short, it depends on a deep theorem of M. Kneser.