47 problems

Let $h(n)$ be the largest $\ell$ such that there is a sequence of primes $p_1<\cdots < p_\ell$ all dividing $n$ with $p_{i+1}\equiv 1\pmod{p_i}$. Let $H(n)$ be the largest $u$ such…

Is it true that, for any $n$, if $d_1<\cdots <d_t$ are the divisors of $n$, then $\sum_{1\leq i<j\leq t}\frac{1}{d_j-d_i} \ll 1+\sum_{1\leq i<t}\frac{1}{d_{i+1}-d_i},$ where the…

Suppose $n = d_1 + ... + d_k$ where the $d_i$ are distinct proper divisors of $n$ but this is not true for any proper divisor of $n$. Must the sum of the reciprocals of all such…

For $n,m\in\mathbb N$, say that $m$ has a medium divisor relative to $n$ if there exists $d\in\mathbb N$ such that $n<d<2n$ and $d\mid m$. For $n,x,y\in\mathbb N$, let … Say that a…

Tenenbaum and I recently asked the following question: let $n_1 < n_2 < \ldots$ be an infinite sequence of positive integers. Is it then true that there always is a positive intege…

Let $A=\{a_1<a_2<\cdots\}$ have positive lower logarithmic density. Must $A$ contain a chain $a_{n_1}\mid a_{n_2}\mid\cdots$ such that …

If $1=d_1<\cdots<d_{\tau(n)}=n$ are the divisors of $n$, then let $\tau_\perp(n)$ count the number of $i$ for which $(d_i,d_{i+1})=1$. Is it true that…

Write the divisors of $n$ as $1=d_1<⋯<d_{τ(n)}=n$. For $α>1$, put $h_α(n)=\sum_{i<τ(n)}(d_{i+1}/d_i-1)^α$. Is $\liminf_{n→∞}h_α(n)<∞$?

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$? Or is this true only for almost…

Let $f\in \mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\geq 1$ for all large $n\in\mathbb{N}$. Does there exist a constant $c=c(f)>0$ such that…

Let $τ(n)$ denote the number of positive divisors of $n$. Is the sequence $τ(n+1)/τ(n)$ dense in $(0,∞)$?

Let $\tau(n)$ denote the number of positive divisors of $n$. Is there an infinite set of natural numbers $n$ such that $\tau(n)=\tau(n+1)$?

Let $F(x)$ be the maximal $k$ such that there exist $n+1,\ldots,n+k\leq x$ with $\tau(n+1),\ldots,\tau(n+k)$ all distinct (where $\tau(m)$ counts the divisors of $m$). Estimate…

If $\tau(n)$ counts the divisors of $n$ then let $f(n)=\sum_{1\leq k\leq n}\tau(2^k-1).$ Does $f(2n)/f(n)$ tend to a limit?

Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+C n^{1/4})$.

Let $\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilon(1)$?

For integer $n\geq 1$ we define the factor difference set of $n$ by $D(n) = \{\lvert a-b\rvert : n=ab\}.$ Is it true that, for every $k\geq 1$, there exist integers…

Let $t\geq 1$ and let $d_t$ be the density of the set of integers $n\in\mathbb{N}$ for which $t$ can be represented as the sum of distinct divisors of $n$. Do there exist constants…

For each $m\in\mathbb N$ and $\alpha\in\mathbb R$, let $\delta(m,\alpha)$ be a density of the set $\{n\in\mathbb N:\exists d,\ d\equiv1\pmod m,\ 1<d<\exp(m^\alpha),\ d\mid n\}$. Is…

Let $k\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\{a_1<a_2<\cdots \}$ be the set of those integers in $[n,n^k]$ which have a divisor in $(n,2n)$. Estimate…

For natural numbers $a,b$, let … Let $\delta_1(a,b)$ be the density of $E(a,b)$. Is it true that, for every function $\delta:\mathbb N\to\mathbb N\to\mathbb R$ such that…

An integer $n$ has been called weird [Ben-Er (74)] if $\frac{\sigma(n)}{n} \geqslant 2$ and $n \neq d_1 + ... + d_k$ where the $d_i$ are distinct proper divisors of $n$. Are there…

For any $n$ let $D_n$ be the set of sums of the shape $d_1,d_1+d_2,d_1+d_2+d_3,\ldots$ where $1<d_1<d_2<\cdots$ are the divisors of $n$. What is the size of…

Let $r(n)$ count divisor pairs $d_1<d_2<2d_1$ of $n$. Is it true that for every $\epsilon>0$, $r(n)<\epsilon\tau(n)$ for almost every $n$?

Define $\tau^+(n)$ to be the number of integers $k$ such that $n$ has a divisor in $[2^k,2^{k+1})$; equivalently, $\tau^+(n)$ is the cardinality of the image of the divisors of $n$…