Carl Pomerance

Let $ω^*(n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p-1$, with $p$ prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of $ω^*(n)$ are still somewhat a mystery. This paper addresses these higher moments and considers other related problems.

Let $Φ(x,y)$ denote the number of integers $n\in[1,x]$ free of prime factors $\le y$. We show that but for a few small cases, $Φ(x,y)<.6x/\log y$ when $y\le\sqrt{x}$.

Given $r \in \mathbb{N}$, define the function $S_{r}: \mathbb{N} \rightarrow \mathbb{Q}$ by $S_{r}(n)=\displaystyle \sum_{k=0}^{n} \frac{k}{k+r} \binom{n}{k}$. In $2015$, the second author conjectured that there are infinitely many $r \in \mathbb{N}$ such that $S_{r}(n)$ is nonintegral for all $n \geq 1$, and proved that $S_{r}(n)$ is not an integer for $r \in \{2,3,4\}$ and for all $n \geq 1$. In $2016$, Florian Luca and the second author raised the stronger conjecture that for any $r \geq 1$, $S_{r}(n)$ is nonintegral for all $n \geq 1$. They proved that $S_{r}(n)$ is nonintegral for $r \in \{5,6\}$ and that $S_{r}(n)$ is not an integer for any $r \geq 2$ and $1 \leq n \leq r-1$. In particular, for all $r \geq 2$, $S_{r}(n)$ is nonintegral for at least $r-1$ values of $n$. In $2018$, the fourth author gave sufficient conditions for the nonintegrality of $S_{r}(n)$ for all $n \geq 1$, and derived an algorithm to sometimes determine such nonintegrality; along the way he proved that $S_{r}(n)$ is nonintegral for $r \in \{7,8,9,10\}$ and for all $n \geq 1$. By improving this algorithm we prove the conjecture for $r\le 22$. Our principal result is that $S_r(n)$ is usually nonintegral in that the upper asymptotic density of the set of integers $n$ with $S_r(n)$ integral decays faster than any fixed power of $r^{-1}$ as $r$ grows.