Roughly squarefree values of the Euler and Carmichael functions
Arithmetic properties of numbers with restricted digits
Nonaliquots and Robbins numbers
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
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
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 . 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
We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if 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 .
On integers with a special divisibility property
In this note, we study those positive integers which are divisible by , where is the Carmichael function.
Consecutive primes in tuples
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 of linear forms in ℤ[x], the set contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that 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 form an increasing (resp....
Representations of integers as sums of primes from a Beatty sequence
On rough and smooth neighbors.
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
We investigate in various ways the representation of a large natural number as a sum of positive -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.
Piatetski-Shapiro sequences
Coincidences in the values of the Euler and Carmichael functions
Concatenations with binary recurrent sequences.
On the congruence .
Arithmetic functions on Beatty sequences
Page 1