Andreas J. Weingartner

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20102021

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math.NT2021

Somewhat smooth numbers in short intervals

Andreas Weingartner

We use exponent pairs to establish the existence of many $x^{a}$ -smooth numbers in short intervals $[x - x^{b}, x]$ , when $a > 1/2$ . In particular, $b = 1 - a - a (1 - a)^{3}$ is admissible. Assuming…

math.NT2020

An extension of the Siegel-Walfisz theorem

Andreas Weingartner

We extend the Siegel-Walfisz theorem to a family of integer sequences that are characterized by constraints on the size of the prime factors. Besides prime powers, this family incl…

math.NT2020

On primes and practical numbers

Carl Pomerance, Andreas Weingartner

A number $n$ is practical if every integer in $[1, n]$ can be expressed as a subset sum of the positive divisors of $n$ . We consider the distribution of practical numbers that are a…

math.NT2019

The constant factor in the asymptotic for practical numbers

Andreas Weingartner

An integer $n \geq 1$ is said to be practical if every natural number $m \leq n$ can be expressed as a sum of distinct positive divisors of $n$ . The number of practical numbers up to…

math.NT2016

A sieve problem and its application

Andreas Weingartner

Let $θ$ be an arithmetic function and let $B$ be the set of positive integers $n = p_{1}^{α_{1}} \dots p_{k}^{α_{k}}$ , which satisfy $p_{j+1} \le θ( p_1^{α_1}\cdots p_{j}^{α_{j}})…

math.NT2010

The distribution functions of $σ (n) / n$ and $n / ϕ (n)$ , II

Andreas Weingartner

Let $σ (n)$ be the sum of the positive divisors of $n$ , and let $A (t)$ be the natural density of the set of positive integers $n$ satisfying $σ (n) / n \geq t$ . We give an improved asym…

Somewhat smooth numbers in short intervals

An extension of the Siegel-Walfisz theorem

On primes and practical numbers

The constant factor in the asymptotic for practical numbers

A sieve problem and its application

The distribution functions of σ(n)/n and n/ϕ(n), II

The distribution functions of $σ (n) / n$ and $n / ϕ (n)$ , II