This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 23

PEN J Problems, 12
Tags: divisor function
Determine all positive integers $n$ such that $n={d(n)}^2$.

PEN J Problems, 20
Tags: modular arithmetic , divisor function
Show that $\sigma (n) -d(m)$ is even for all positive integers $m$ and $n$ where $m$ is the largest odd divisor of $n$.

PEN J Problems, 19
Tags: divisor function
Prove that $\sigma(n)\phi(n) < n^2$, but that there is a positive constant $c$ such that $\sigma(n)\phi(n) \ge c n^2$ holds for all positive integers $n$.

PEN J Problems, 1
Tags: divisor function
Let $n$ be an integer with $n \ge 2$. Show that $\phi(2^{n}-1)$ is divisible by $n$.

2024 IRN-SGP-TWN Friendly Math Competition, 2
Tags: number theory , divisor function
Let $d(n)$ denote the number of positive divisors of $n$. For any given integer $a \geq 3$, define a sequence $\{a_i\}_{i=0}^\infty$ satisfying [list] [*] $a_{0}=a$, and [*] $a_{n+1}=a_{n}+(-1)^{n} d(a_{n})$ for each integer $n \geq 0$. [/list] For example, if $a=275$, the sequence would be \[275, \overline{281,279,285,277,279,273}.\] Prove that for each positive integer $k$ there exists a positive integer $N$ such that if such a sequence has period $2k$ and all terms of the sequence are greater than $N$ then all terms of the sequence have the same parity. [i]Proposed by Navid[/i]

2020 Baltic Way, 19
Tags: number theory , divisor function
Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Prove that there are infinitely many positive integers $n$ such that $\left\lfloor\sqrt{3}\cdot d(n)\right\rfloor$ divides $n$.

PEN J Problems, 5
Tags: floor function , divisor function , relatively prime , number theory
If $n$ is composite, prove that $\phi(n) \le n- \sqrt{n}$.

PEN J Problems, 6
Tags: prime factorization , modular arithmetic , divisor function , quadratic , relatively prime , number theory
Show that if $m$ and $n$ are relatively prime positive integers, then $\phi( 5^m -1) \neq 5^{n}-1$.

PEN J Problems, 2
Tags: abstract algebra , real analysis , divisor function , group theory , function
Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]

PEN J Problems, 16
Tags: divisor function
We say that an integer $m \ge 1$ is super-abundant if \[\frac{\sigma(m)}{m}>\frac{\sigma(k)}{k}\] for all $k \in \{1, 2,\cdots, m-1 \}$. Prove that there exists an infinite number of super-abundant numbers.

PEN J Problems, 9
Tags: divisor function
Show that the set of all numbers $\frac{\phi(n+1)}{\phi(n)}$ is dense in the set of all positive reals.

PEN J Problems, 21
Tags: divisor function
Show that for any positive integer $n$, \[\frac{\sigma(n!)}{n!}\ge \sum_{k=1}^{n}\frac{1}{k}.\]

PEN J Problems, 14
Tags: divisor function
Find all positive integers $n$ such that ${d(n)}^{3} =4n$.

PEN J Problems, 3
Tags: number theory , modular arithmetic , greatest common divisor , divisor function
If $p$ is a prime and $n$ an integer such that $1<n \le p$, then \[\phi \left( \sum_{k=0}^{p-1}n^{k}\right) \equiv 0 \; \pmod{p}.\]

PEN J Problems, 4
Tags: algebra , arithmetic sequence , divisor function , polynomial
Let $m$, $n$ be positive integers. Prove that, for some positive integer $a$, each of $\phi(a)$, $\phi(a+1)$, $\cdots$, $\phi(a+n)$ is a multiple of $m$.

PEN J Problems, 15
Tags: divisor function
Determine all positive integers for which $d(n)=\frac{n}{3}$ holds.

PEN J Problems, 17
Tags: function , totient function , divisor function , number theory
Show that $\phi(n)+\sigma(n) \ge 2n$ for all positive integers $n$.

PEN J Problems, 18
Tags: divisor function
Prove that for any $\delta$ greater than 1 and any positive number $\epsilon$, there is an $n$ such that $\left \vert \frac{\sigma (n)}{n} -\delta \right \vert < \epsilon$.

PEN J Problems, 7
Tags: divisor function
Show that if the equation $\phi(x)=n$ has one solution, it always has a second solution, $n$ being given and $x$ being the unknown.

PEN J Problems, 10
Tags: inequalities , divisor function
Show that [list=a] [*] if $n>49$, then there are positive integers $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}<1$. [*] if $n>4$, then there are $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}>1$.[/list]

PEN J Problems, 8
Tags: divisor function , logarithm
Prove that for any $ \delta\in[0,1]$ and any $ \varepsilon>0$, there is an $ n\in\mathbb{N}$ such that $ \left |\frac{\phi (n)}{n}-\delta\right| <\varepsilon$.

PEN J Problems, 22
Tags: inequalities , divisor function
Let $n$ be an odd positive integer. Prove that $\sigma(n)^3 <n^4$.

PEN J Problems, 13
Tags: divisor function
Determine all positive integers $k$ such that \[\frac{d(n^{2})}{d(n)}= k\] for some $n \in \mathbb{N}$.