O číslech spřízněných a dokonalých. [II.]
On 4/n = 1/x + 1/y + 1/z and Rosser's sieve
On a Brun-Titchmarsh inequality for multiplicative functions
On a density problem of Erdös
On a ternary quadratic form over primes
On a theorem of Ramachandra
On a thin set of integers involving the largest prime factor function.
On characters of order p (mod p²)
On consecutive integers divisible by the number of their divisors
We prove that there are no strings of three consecutive integers each divisible by the number of its divisors, and we give an estimate for the number of positive integers n ≤ x such that each of n and n + 1 is a multiple of the number of its divisors.
On Divisor Problems in Connection with Squarefree Numbers.
On divisors whose sum is a square
On extremal sets without coprimes
On generalized square-full numbers in an arithmetic progression
Let and . Denote by the set of all integers whose canonical prime representation has all exponents
On ideals free of large prime factors
In 1989, E. Saias established an asymptotic formula for with a very good error term, valid for , , We extend this result to an algebraic number field by obtaining an asymptotic formula for the analogous function with the same error term and valid in the same region. Our main objective is to compare the formulae for and and in particular to compare the second term in the two expansions.
On positive integers with a certain nondivisibility property.
On prime divisors of Mersenne numbers
On prime factors of integers of the form (ab+1)(bc+1)(ca+1)
1. Introduction. For any integer n > 1 let P(n) denote the greatest prime factor of n. Győry, Sárközy and Stewart [5] conjectured that if a, b and c are pairwise distinct positive integers then (1) P((ab+1)(bc+1)(ca+1)) tends to infinity as max(a,b,c) → ∞. In this paper we confirm this conjecture in the special case when at least one of the numbers a, b, c, a/b, b/c, c/a has bounded prime factors. We prove our result in a quantitative form by showing that if is a finite set of triples (a,b,c)...
On rough and smooth neighbors.
We study the behavior of the arithmetic functions defined byF(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.
On smooth integers in short intervals under the Riemann Hypothesis
On Some Connections Between Zeta-Zeros and Square-Free Integers.