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 $x \rightarrow \infty$

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

Integers with dense divisors

AbstractLet 1=d1(n)<d2(n)<⋯<dτ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max1⩽i<τ(n)di+1(n)/di(n)⩽y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let u=logx/logy, and v=logx/logz. We show that x−1D(x,y,z)logz is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)≍(1−u/v)/(u+1) for v⩾3+ε. Finally, we show that d(u,v)=e−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x−1D(x,y,1), for fixed u. This leads to a new estimate for d(u)

The limiting distribution of the divisor function

AbstractLet σα(n) be the sum of the αth power of the positive divisors of n. We establish an asymptotic formula for the natural density of the set of integers n that satisfy σα(n)/nα⩾t, as t→∞. Two other limiting distributions considered are based on Jordan's totient function and Dedekind's psi function