# Questions tagged [divisors-multiples]

For questions on divisors and multiples, mainly but not exclusively of integers, and related and derived notions such as sums of divisors, perfect numbers and so on.

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### Bounds of heights of coefficients of rational polynomials

For a non zero rational $r=p/q$ ($p,q\in\mathbb Z$ coprimes), define the height of $r$ by $\mathrm{ht}(r)=\max(|p|,|q|)$ (by convention $\mathrm{ht}(0)=0$). For a polynomial $P\in\mathbb Q[X]$, define ...
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### Convolution sum of divisor functions

Let $\sigma_0(n)$ be the divisor counting function $$\sigma_0(n) = \sum_{d \vert n} 1.$$ I'm interested in the convolution sum $$S(n) := \sum_{k=1}^{n-1} \sigma_0(k) \sigma_0(n-k)$$ I ran some quick ...
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### Is the divisor counting function equidistributed mod $p$?

Let $\sigma_0(n)$ be the divisor counting function: $$\sigma_0(n) = \sum_{d \vert n} 1.$$ I ran some numerical experiments that showed when $p$ is prime, the function $\sigma_0(n)$ is equidistributed ...
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### On "Euclidean" odd perfect numbers

In what follows, we let $N = r^s u^2$ be an odd perfect number given in Eulerian form, i.e. $r$ is the special prime satisfying $r \equiv s \equiv 1 \pmod 4$ and $\gcd(r,u)=1$. In this preprint, ...
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### On Carmichael function and aliquot parts of odd perfect numbers

I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
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### Divisor of given order in short intervals

Is the following Open question or Conjecture already known, or eventually settled ? Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
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### Divisibility of Stirling numbers

It is well known that if $p$ is prime, Stirling numbers of the first and second kind, $s_1(p,k)$ and $s_2(p,k)$, are divisible by $p$ if $1<k\le p-1$ (Lagrange ; easiest is working in $\mathbb F_p$ ...
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### Greatest common divisors of some binomial coefficients

This is cross-posted from math.stackexchange. While making some computation, I stumbled upon a curious relation among some binomial coefficients. Consider the sequence of binomial coefficients $a(k,n)$...
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### When an element of a ring that is divisible by a finite set of elements is necessarily divisible by their product?

In a commutative ring $R$, when does the assumption $r_i\mid r$ for $1\le i\le n$ imply $\prod_{1\le i\le n} r_i\mid r$ (when $r_i$ are fixed)? Does there exist any criterion for this implication that ...
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### Divisibility relation with a specific sum of divisors

Let $\sigma(n)$ be the sum of the divisors of $n$. Is it always true that if $n$ is odd, that $$n\mid\sum_{k=1}^{\frac{n-1}{2}}k^2\sigma(k)\sigma(n-k)?$$ I have checked this up to $n=100$, and I ...
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### Shapiro inequality for divisor sets

The Shapiro inequality is the statement that if $x_1, x_2, \dots, x_n$ are positive, with $x_{n+1}=x_1, x_{n+2}=x_2$, then $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}.$$ This can be ...
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### Divisibility chains and polynomials

Let $P\in \Bbb{Z}[X]$ be a polynomial with degree $d>1$. It is conjectured that for all such $P$, their range for integer inputs $R_P:=P(\Bbb{Z})$ has finite intersection with the set of factorials ...
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### On the diophantine equation $x^{m-1}(x+1)=y^{n-1}(y+1)$ with $x>y$, over integers greater or equal than two

I've asked two years ago a post on Mathematics Stack Exchange, were provided two excellent answers. I'm asking on MathOverflow in the hope that some professor can to expand/improve (if it is possible) ...
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### Maximal number of divisors of numbers whose sum does not exceed $n$

Denote by $f(n)$ the maximal number of distinct divisors of $k$ integer numbers $1\leq a_1<a_2<\ldots<a_k\leq n$, where $k$ is not fixed and $a_1+\ldots+a_k\leq n$. I'm interested in the ...
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### Arithmetic billiards, prime numbers and the Goldbach conjecture

I've edited the following post on Mathematics Stack Exchange, (now closed, at that date I'm suspended) with identifier 4510963, please let me to know if you've some doubt or I can improve the post. On ...
1 vote
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### Number of distinct near-squares primes dividing an odd perfect number

I'm curious about if the following question is in the literature or what work can be done about it. Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function ...
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### If $p^k m^2$ is an odd perfect number with special prime $p$, then under what other conditions on $\sigma(p^k)/2$ does $k=1$ follow?

Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Descartes (1638), Frenicle (1657), and subsequently [Sorli (2003) - ...
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### A diophantine equation inspired in a conjecture due to Gica and Luca, example of a large Mersenne exponent

In this post I consider the equation $$k\cdot x=y^2+z^2(x^2-2)-2\tag{1}$$ over odd integers $y\geq 1$ and $z\geq 1$, and over integers $k\geq 1$ and very large Mersenne exponents $x$ such that $x^2-2$ ...
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### Small covering of divisors

Let $D_n$ be the set of divisors of $n$. Does there always exists a $B\subseteq D_n$ such that $D_n = \{\gcd(ab,n) \mid a\leq \sqrt{n}, b\in B\}$ and $\sum_{b\in B} \frac{n}{b}=O(n)$?
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### A conjecture concerning the equation $\sigma\left(\square\right)=\text{prime}$

I can deduce the following simple proposition, the definitions for $\sigma(x)$ the sum of divisors functions and $\varphi(x)$ the Euler totient function are assumed. After I present a conjecture that ...
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### A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
### Around the equation $\sigma\left(\square\right)=\text{prime}$: counterexamples or a proof for some of these conjectures
For integers $A,B\geq 1$ we define the difference $\sigma(A)\sigma(B)-\sigma(AB)$, denoting it as $[A,B]$, where $\sigma(n)=\sum_{1\leq d\mid n}d$ denotes the sum of divisors function. It is possible ...