Found problems: 35
For a positive integer $n$ we denote by $d(n)$ the number of positive divisors of $n$ and by $s(n)$ the sum of these divisors. For example, $d(2018)$ is equal to $4$ since $2018$ has four divisors $(1, 2, 1009, 2018)$ and $s(2018) = 1 + 2 + 1009 + 2018 = 3030$.
Determine all positive integers $x$ such that $s(x) \cdot d(x) = 96$.
(Richard Henner)
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$.
Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.
For a positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$. Determine all positive integers $n$ for which $d(n)$ is the second largest divisor of $n$.
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that
$$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$
for all $n\ge 1$
[i]Cyprus[/i]
For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$.
For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$
[i]Proposed by Suhaimi Ramly, Malaysia[/i]
The positive integers $a $ and $b $ satisfy the sistem $\begin {cases} a_{10} +b_{10} = a \\a_{11}+b_{11 }=b \end {cases} $ where $ a_1 <a_2 <\dots $ and $ b_1 <b_2 <\dots $ are the positive divisors of $a $ and $b$ .
Find $a$ and $b $ .
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that
$$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$
for all $n\ge 1$
[i]Cyprus[/i]