# 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.

**49**

votes

**4**answers

4k views

### Strange (or stupid) arithmetic derivation

Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...

**49**

votes

**1**answer

2k views

### Iterations of 2^(n-1)+5: the strong law of small numbers, or something bigger?

I've discovered what I believe is a quite remarkable sequence (A318970), defined by
$$n_1 = 3,\qquad n_{k+1} = 2^{n_k-1}+5\quad(k\geq 1).$$
Here are the first four terms with their prime ...

**34**

votes

**2**answers

1k views

### Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...

**27**

votes

**2**answers

5k views

### Why do primes dislike dividing the sum of all the preceding primes?

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...

**15**

votes

**3**answers

2k views

### Are there pairs of consecutive integers with the same sum of factors?

Background/Motivation
I was planning to explain Ruth-Aaron pairs to my son, but it took me a few moments to remember the definition. Along the way, I thought of the mis-definition, a pair of ...

**14**

votes

**0**answers

363 views

### Do the coefficients of these irreducible polynomials always become periodic?

Fix $n\in\mathbb N$ and a starting polynomial (or seed) $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_0a_n\ne0$.
Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+...

**13**

votes

**2**answers

630 views

### Number of distinct factors

Denote $\omega(m)$ to be number of distinct factors of $m$ as defined in http://mathworld.wolfram.com/DistinctPrimeFactors.html.
At every $c>0$, given $n\in\Bbb N$ define $$S(n,c)=\big\{m\in\Bbb N:...

**12**

votes

**2**answers

562 views

### Find all $m$ such $2^m+1\mid5^m-1$

The problem comes from a problem I encountered when I wrote the article
Find all positive integer $m$ such
$$2^{m}+1\mid5^m-1$$
it seem there no solution. I think it might be necessary to use ...

**12**

votes

**3**answers

662 views

### On the number of consecutive divisors of an integer

Define for $n \in \mathbb{N}$ the function $$\tau_1(n):=\sum_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau_1(n)$ is $1$ since $$\sum_{n\leq ...

**11**

votes

**3**answers

2k views

### On Robin's criterion for RH [closed]

\begin{equation}
\sigma(n) < e^\gamma n \log \log n
\end{equation}
In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984)....

**11**

votes

**2**answers

818 views

### Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted from MSE.)
Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...

**11**

votes

**2**answers

1k views

### Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k$?

This question related to this question from SE. I'm interested to know if there exists an integer $x>1$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$
for all positive integers $k$.
Note. $\...

**10**

votes

**1**answer

337 views

### Laurent polynomials associated to partitions and a $Q$-deformation of $\sigma(d)$

Let $\alpha \vdash d$ be a partition of $d$, i.e. $\alpha = (\alpha_1 \geq \alpha_2 \geq …\geq \alpha_l)$, where $\sum_k \alpha_k = d$. Define a Laurent polynomial in $Q$ as follows:
$$
P_\alpha(Q) = ...

**9**

votes

**2**answers

888 views

### runs of consecutive non squarefree integers

This question gained no attention at Math SE.
Call a sequence of $k$ consecutive naturals squary if each one of them is divided by a square > 1. The Chinese Remainder theorem trivially guarantees us ...

**8**

votes

**2**answers

1k views

### Does the Prime Number Theorem have anything to do with Erdos-Kac law or vice versa?

The prime number theorem says on average we can find $\frac n{\log n}$ primes of magnitude $n$.
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ primes.
Somehow the fact $e^{\...

**8**

votes

**1**answer

293 views

### Sum of divisors below threshold

Let $\sigma(n)$ denote the sum of divisors of $n$, that is,
$$
\sigma(n) = \sum_{d | n} d.
$$
It is known that $\sigma$ can have values as large as order $n \log \log n$. However, obviously the sum is ...

**8**

votes

**1**answer

642 views

### Short divisor sum

Let $d(n)$ denote the number of positive divisors of the positive integer $n$.
Pick some positive $X,h \in \mathbb{R}$ and consider the sum
$$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$
In view of ...

**7**

votes

**6**answers

1k views

### Getting a bound on the coefficients of the factor polynomial

Suppose $f(x):=a_0+a_1x+\cdots+a_nx^n$ is a polynomial in $\mathbb{Z}[x]$ and $|a_i|\leq M$ for each $i=0,\ldots ,n.$ Now suppose $g(x)$ is a factor of $f(x)$ in $\mathbb{Z}[x]$, then is it possible ...

**7**

votes

**2**answers

1k views

### Numbers with all N-digit prefixes divisible by N

In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. ...

**7**

votes

**2**answers

969 views

### Smallest integer not divisible by integers in a finite set

Hello all, if $a_1,a_2, \ldots a_t$ are $t$ integers $\geq 2$, the set
$G(a_1,a_2, \ldots a_t)=\lbrace N \geq 1 |$ In any sequence of $N$ consecutive
integers there is at least one not divisible by ...

**7**

votes

**0**answers

451 views

### When is $ \sigma(n!-1) $ a perfect square?

I am looking for pairs of positive integers $(m,n)$ such that $ \sigma(n!-1) =m^2$, where $\sigma$ is the sum of divisors function. Examples occur with $(m,n)=(12,5),(1,2)$.
Question: Are there ...

**6**

votes

**1**answer

266 views

### Is $\liminf \frac{\sigma_{k}(n)}{n}$ finite for every $k$?

Can someone show me how to prove that $$\liminf_{n \to \infty} \frac{\sigma_{k}(n)}{n} < \infty$$ for every natural number $k$? Or is this problem open?
Here, $\sigma_{k}(n)=\sigma(\sigma(\sigma(\...

**6**

votes

**1**answer

234 views

### Large gaps between consecutive irreducible polynomials with small heights

For a prime gap of length at least $n$, a trivial upper bound for its first occurrence is $N=n!$ or $N=lcm(2,\dots,n)$. A bit better is $N=p_1\cdots p_n$ where $p_k$ is the $k$th prime, as then $N+2,\...

**6**

votes

**2**answers

968 views

### Sum of divisor function over arithmetic progression

I am trying to find an estimate for the following sum:
$$
\sum_{\substack{n \leq x \\ n \equiv k (m)}} d(n),
$$
where $d(n)$ is number of divisors of $n$. I found estimates for the case when $k$ and ...

**6**

votes

**0**answers

151 views

### The total number of divisors of those integers with the most divisors

I'm interested in summing $\tau(m)$, the number of positive divisors of $m$, not over all integers in an interval but rather over only the integers with the most divisors. More specifically:
Given a ...

**5**

votes

**1**answer

621 views

### Arbitrarily large $n$ divides $F_n$

Is it true that there exists $n \in \mathbb{N}$ with arbitrarily many prime factors such that $n$ divides $F_n$, where $F_n$ represents the n-th Fibonacci number?

**5**

votes

**1**answer

116 views

### How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free?

I am interested in obtaining an upper bound for $\prod_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that
$$
\prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...

**5**

votes

**1**answer

456 views

### Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?

I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also ...

**5**

votes

**1**answer

224 views

### Friable Numbers In Short Intervals: Density Estimates?

I am hoping for explicit numerical estimates like the following sample (with made up numbers, though it might be true): for every $n \gt 10^6$ and every $b$ with $b^2 \lt n \lt b^3$, the number of ...

**4**

votes

**3**answers

300 views

### Good books on the divisor sum function $\sigma(n)$?

I would like gain detailed knowledge about properties of the divisor sum function $\sigma(n)$, special equation that have been studied (e.g. $\sigma(n) = 2n$ perfect numbers, ...) and progress that ...

**4**

votes

**1**answer

214 views

### Improvement of a bound on divisor distributions from “Divisors” (Hall and Tenenbaum)?

In the classic text referred to in the title of this question, the bound
$$
H(x,y,2y) \ll \frac{x}{(\log y)^{\delta}\sqrt{\log \log y}},\quad (3\leq y\leq \sqrt{x})
$$
is given, where $\delta=1-\frac{...

**4**

votes

**2**answers

672 views

### What literature is known about MacMahon's generalized sum-of-divisors function?

MacMahon in the paper Divisors of Numbers and their Continuations in the Theory of Partitions defines several generalized notions of the sum-of-divisors function; for example, if we write $a_{n,k}$ ...

**4**

votes

**0**answers

580 views

### The Grimm Machine(s): A Collatz Conjecture Rival?

Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
Just as the Collatz ...

**4**

votes

**0**answers

295 views

### Maximal order of Hooley's Delta function?

There is a large literature on Hooley's
$$
\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1
$$
giving its normal and average order. What is known of its maximal order?
Clearly $\Delta(n)\le d(n)$ ...

**3**

votes

**1**answer

292 views

### Primitive sequence $a_i$ attaining Pillai's bound on $\sum_{i} 1/a_i$

A primitive sequence $1<a_1<\ldots<a_k\leq n$ is a sequence of integers no one of which divides any other, investigated by Erdos, Behrend and others, over the last 80 years. In fact, $\max k=\...

**3**

votes

**2**answers

201 views

### Sum of small divisors with powers

I am looking for the tightest known bound for the sum
$$\sum_{\substack{1\leq k\leq j^\alpha \\ k\mid j}}k^\lambda$$
where $j$ is a large positive integer, $\alpha\in(0,1)$ and $\lambda\geq 1$.
I ...

**3**

votes

**1**answer

220 views

### Divisibility properties of a stream of numbers

Let $S$ be a set of $k$ distinct natural numbers, each
from the interval $[2,n]$, with least common multiple $\mathop{lcm}$.
What fraction $\rho$ of the numbers $2,3,4,\ldots,\mathop{lcm}$
are ...

**3**

votes

**2**answers

2k views

### Finding primes using Euler's sum of divisors recurrence relation

Euler came up with following recurrence relation for the sum of divisors (refer to http://arxiv.org/abs/math/0411587)
$$\sigma(n) = \sigma(n−1) + \sigma(n−2) − \sigma(n−5) − \sigma(n−7) \dots$$
Since ...

**3**

votes

**1**answer

338 views

### Ratio of consecutive divisors and average

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:
(1) ...

**3**

votes

**1**answer

1k views

### If $N = qn^2$ is an odd perfect number with $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$?

The title says it all.
Question
If $N = qn^2$ is an odd perfect number with Euler prime $q$ and $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$?
Heuristic
From the Descartes spoof, ...

**3**

votes

**1**answer

286 views

### Problem related to Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.
Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ good if,
for any $r,s,u,...

**3**

votes

**0**answers

117 views

### Is there a way to reduce this problem to two variables through functions coming from arithmetic?

Consider following diophantine equation in $\mathbb Z[x,y,z]$ in three integer variables $x,y,z$
$$x^2+L(y,z)x+L_1(y)L_2(z)=0$$ where $L(y,z)$ is a non-homogeneous linear polynomial in $y,z$ and $L_1(...

**3**

votes

**1**answer

350 views

### If $q^k n^2$ is an odd perfect number with Euler prime $q$, is $\sigma(q^k)/n + \sigma(n)/q^k$ bounded from above?

STATEMENT OF THE PROBLEM
If $q^k n^2$ is an odd perfect number with Euler prime $q$, is $\sigma(q^k)/n + \sigma(n)/q^k$ bounded from above?
MOTIVATION
Let $\sigma=\sigma_{1}$ denote the classical ...

**3**

votes

**0**answers

199 views

### Any counter example for this: ${\phi(2^n-1)} \bmod \tau(2^n-1)=0$ for every integer $n \geq 1$? [closed]

I asked this question here In S.E but i don't received any resposnes for it, I would like to know if it is appropriate for M.O.
I'm always interesting for properties of the following series : $ \...

**2**

votes

**1**answer

182 views

### Piltz Divisor Problem

Let $\tau_k(n)$ count the number of ways of representing $n$ as the product of $k$ natural numbers. It is known that:
$$D_k(x) = \sum_{n \leq x} \tau_k(n) = xP_k(\log x) + O(x ^{1 - \frac{1}{k-1}}(\...

**2**

votes

**1**answer

189 views

### Minimal $n$ such that $(a-1)^m | a^n - 1$ for a given $a,m > 1$

This open-ended question was originally posted on Twitter here. Specifically,
Problem
Given $a,m \in \mathbb{N}$ with $a, m \gt 1$, find the minimal value $n \in \mathbb{N}$ such that $(a-1)^m \mid ...

**2**

votes

**2**answers

604 views

### Is it necessary that gcd > 1 of an infinite set? [closed]

Consider an infinite set $S$, of positive integers.
If all the finite subsets of $S$ have GCD $>$ $1$, is it necessary that the GCD of $S$ is greater than $1$ as well?

**2**

votes

**1**answer

351 views

### Trying to prove a congruence for Stirling numbers of the second kind

This a repost of a question I asked at Stack Exchange, but I got no answer so far, so I am trying here, even though it may not suit the "research level" requirement.
Proposition: When $n$ and $m$ are ...

**2**

votes

**1**answer

179 views

### A truncated divisor sum

I am interested in an upper bound for
$$\sum_{\substack{d\mid N\\ d>A}}\frac{1}{d^3},$$
in particular, I can show that above is
$$\ll\frac{\text{exp}\left(C\frac{\log(N)}{\log\log(N)}\right)}{A^...

**2**

votes

**1**answer

203 views

### Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that
$$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n a_{i}^{...