arXiv subjects
Publications and source records attributed to Jean-Marie De Koninck.
We study the set of integers $n \le x$ for which $\sum_{i=1}^{n}\frac{1}{\gcd(n,i)}$ takes an integer value. We show that the number of such integers is $\ll x^{2/3+ε}$ for all fixed $ε> 0$.
A sequence of integers $1=a_0 m^{0.9}$. Moreover, denoting by $G(m,r)$ the number of \emph{distinct} addition chains of length $m+r$ leading to an integer $n\in [2^m, 2^{m+1})$, we show that there exist positive constants $K_3$ and $K_4$ such that $$ K_3^r \left(\frac{m^2}{r}\right)^r\le G\left(m,r\right)\le K_4^r \left(\frac{m^2}{r} \right)^r $$ provided $m^{0.9}<r<m$. This improves and generalizes previous results on the minimal length of addition chains and addresses a question raised by Paul Erdős.
We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$ and we define the counting function $$ F(m,r):=\#\left\{n\in[2^m, 2^{m+1}):\ell(n)\le m+r\right\}, $$ where $m$ is a positive integer and $r\ge 0$ is a real number. We show that for $0< c<\log 2$ and for any $\varepsilon>0$, we have as $m\to \infty$, $$ F\left(m,\frac{cm}{\log m}\right)<\exp\left(cm+\frac{\varepsilon m\log\log m}{\log m}\right) $$ and $$ F\left(m,\frac{cm}{\log m}\right)>\exp\left(cm-\frac{(1+\varepsilon)cm\log\log m}{\log m}\right). $$ This extends a result of Erdős which says that for almost all $n$, as $n\to\infty$, $$ \ell(n)=\frac{\log n}{\log 2}+\left(1+o(1)\right)\frac{\log n}{\log \log n}. $$
For various arithmetic functions $f:\mathbb{N} \to \mathbb{R}$, the behavior of $f(n!)$ and that of $\sum_{n\le N} f(n!)$ can be intriguing. For instance, for some functions $f$, we have ${f(n!)=\sum_{k\le n}f(k)}$, for others, we have ${f(n!)=\sum_{p\le n}f(p)}$ (where the sum runs over all the primes $p\le n$). Also, for some $f$, their minimum order coincides with $\lim_{n\to \infty}f(n!)$, for others, it is their maximum order that does so. Here, we elucidate such phenomena and more generally, we embark on a study of $f(n!)$ and of $\sum_{n\le N}f(n!)$ for a wide variety of arithmetical functions $f$. In particular, letting $d(n)$ and $σ(n)$ stand respectively for the number of positive divisors of $n$ and the sum of the positive divisors of $n$, we obtain new accurate asymptotic expansions for $d(n!)$ and $σ(n!)$. Furthermore, setting $ρ_1(n):=\max\{d\mid n:d\le \sqrt n\}$ and observing that no one has yet obtained an asymptotic value for $\sum_{n\le N} ρ_1(n)$ as $N\to \infty$, we show how one can obtain the asymptotic value of $\sum_{n\le N} ρ_1(n!)$.
Under the fundamental theorem of arithmetic, any integer $n>1$ can be uniquely written as a product of prime powers $p^a$; factoring each exponent $a$ as a product of prime powers $q^b$, and so on, one will obtain what is called the tower factorization of $n$. Here, given an integer $n>1$, we study its height $h(n)$, that is, the number of "floors" in its tower factorization. In particular, given a fixed integer $k\geq 1$, we provide a formula for the density of the set of integers $n$ with $h(n)=k$. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights.
A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$.
Let $τ(n)$ stand for the number of divisors of the positive integer $n$. We obtain upper bounds for $τ(n)$ in terms of $\log n$ and the number of distinct prime factors of $n$.