Practical numbers and the distribution of divisors

An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$. We show that the number of practical numbers below $x$ is asymptotic to $c x/\log x$, as conjectured by Margenstern. We also give an asymptotic estimate for the number of integers below $x$ whose maximum ratio of consecutive divisors is at most $t$, valid uniformly for $t\ge 2$.Comment: Minor improvement of Theorems 2 and 4. To appear in Q. J. Mat

On the degrees of polynomial divisors over finite fields

We show that the proportion of polynomials of degree $n$ over the finite field with $q$ elements, which have a divisor of every degree below $n$, is given by $c_q n^{-1} + O(n^{-2})$. More generally, we give an asymptotic formula for the proportion of polynomials, whose set of degrees of divisors has no gaps of size greater than $m$. To that end, we first derive an improved estimate for the proportion of polynomials of degree $n$, all of whose non-constant divisors have degree greater than $m$. In the limit as $q \to \infty$, these results coincide with corresponding estimates related to the cycle structure of permutations.Comment: 21 pages, 1 figur

On integers nn for which Xn−1X^n-1 has a divisor of every degree

A positive integer $n$ is called $\varphi$-practical if the polynomial $X^n-1$ has a divisor in $\mathbb{Z}[X]$ of every degree up to $n$. In this paper, we show that the count of $\varphi$-practical numbers in $[1, x]$ is asymptotic to $C x/\log x$ for some positive constant $C$ as $x \rightarrow \infty$

Black Carbon Contribution to the Aerosol Phase and its Scavenged Fraction in Mixed Phase Clouds at the High Alpine Site Jungfraujoch (3580m asl)

The mass fraction of black carbon (BC) in the atmospheric aerosol and its mixing state are important for the direct aerosol climate effect. These properties also determine if BC is incorporated into cloud hydrometeors (i.e. droplets and ice crystals) and are important because the microphysical and optical properties of the cloud are altered (indirect aerosol effect). Measurements were performed during several Cloud and Aerosol Characterization Experiments, in winter 2004 (CLACE3), summer 2004 (CLACE3.5), winter 2005 (CLACE4) and summer 2005 (CLACE4.5) at the high Alpine research station Jungfraujoch (3580 m asl)

Uniform distribution of αn\alpha n modulo one for a family of integer sequences

We show that the sequence $(\alpha n)_{n\in \mathcal{B}}$ is uniformly distributed modulo 1, for every irrational $\alpha$, provided $\mathcal{B}$ belongs to a certain family of integer sequences, which includes the prime, almost prime, squarefree, practical, densely divisible and lexicographical numbers. We also give an estimate for the discrepancy if $\alpha$ has finite irrationality measure.Comment: 9 page

An Erd\H{o}s-Kac theorem for integers with dense divisors

We show that for large integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$, where $C=1/(1-e^{-\gamma})\approx 2.280$ and $V\approx 0.414$. This result is then generalized in two different directions.Comment: 28 page