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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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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^{\...
2
votes
1answer
183 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
1answer
180 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^...
3
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1answer
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 ...
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0answers
129 views

On sets of coprime integers in intervals

Briefly, Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval? The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
2
votes
1answer
399 views

On comparing two almost injective divisor maps

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 In an introductory post on ...
12
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2answers
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 ...
8
votes
1answer
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 ...
49
votes
1answer
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 ...
1
vote
1answer
283 views

Does there exist an integer that is both solitary and almost perfect?

This question is an offshoot from the following MSE post. I hope that it is appropriate for this site. Let $\sigma(x)$ be the sum of the divisors of $x$. An integer $a$ is said to be solitary if ...
0
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0answers
91 views

gcd of polynomial values

Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...
2
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1answer
128 views

The number of numbers no greater than n that are divisible by all their suffixes

My question: what a formula for finding the number of numbers no greater than n that are divisible by all their suffixes. e.g: 5, 25, 125, 0125, 70125 are divisors of 70125. refinement: $\overline{0....
3
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0answers
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(...
2
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0answers
173 views

A conjecture regarding odd perfect numbers

(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.) Let $\sigma(z)$ denote the sum of ...
11
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3answers
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)....
0
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1answer
152 views

$2$-adic valuations and sum of divisor function

Consider the sum of $k^{th}$-power of divisors of $n$, denoted $$\sigma_k(n)=\sum_{d\vert n}d^k.$$ Let $\nu_p(x)$ stand for the $p$-adic valuation of the integer $x$. The following appears to be ...
4
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0answers
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 ...
3
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1answer
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, ...
1
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1answer
449 views

Some divisibility constraints in 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 $\mathsf{gcd}(a,b)=1$ excellent if linear form ...
8
votes
1answer
295 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 ...
6
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0answers
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 ...
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vote
0answers
215 views

Prove A Skipping Prime Conjecture For Rio?

I am writing a paper to accompany a Short Communication I plan to give in Rio this August. The paper regards work on jumping primes, a project on which Jose Brox has been working with me. I was going ...
1
vote
0answers
81 views

Existence of equation about the product of the divisor sum function

Let $\sigma_k(n)$ be the sum of the $k$-th powers of the positive divisors of $n$ and $\mu(n)$ be the Möbius function. As Arithmetic function - Wikipedia mentioned, there is an equation that $$\...
3
votes
2answers
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
1answer
105 views

How much information is required to determine integers x,y,z [closed]

what is x+y+z is x,y and z are integers and xy-1 is divisible by z, yz-1 is divisible by x and xz-1 is divisible by y.
4
votes
3answers
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 ...
2
votes
1answer
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 ...
8
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2answers
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^{\...
0
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0answers
90 views

Eigenvalues of a sequence of matrices involving the divisor function

Let $A_{n,k},k=1,\ldots,n$ be a sequence of $n\times n$ upper triangular matrices where $A_{n,1}=I_n$ and $A_{n,k},\quad 2\leq k\leq n$ be a regularly shifted and scaled matrix, with $P_{n,k}$ an $n\...
5
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1answer
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 ...
1
vote
1answer
185 views

Generalized notion of divisor function?

Divisor function $d(n,m)$ counts the number of $q\in\Bbb N$ with $b<q<m$ such that $n\bmod q\equiv0$. Given $b>0$ what is the correct asymptotic, probabilistic and average case behavior of ...
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0answers
125 views

Is there an integer $r \neq q$ (with $r>1$) such that $N = q^k n^2 = \frac{r(r+1)}{2}\cdot{d}$ is an odd perfect number with $d>1$?

Slowak showed in 1999 that every odd perfect number $N = q^k n^2$ can be written in the form $$N = \dfrac{{q^k}\sigma(q^k)}{2}\cdot{D}$$ where $D>1$. From this result, it follows that every odd ...
2
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1answer
208 views

On attempting a proof for $r > 1$, if $M = {2^r}{b^2}$ is an even almost perfect number which is not a power of two

(Preamble: I first thought that this question might be more appropriate for MSE. However, I posted it here nonetheless in the hope that someone with that brilliant idea can help with answering my ...
11
votes
2answers
819 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 ...
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0answers
78 views

An arithmetic function involving arbitrary (but fixed) number of divisors

I need at least basic information about generating functions of the following class of arithmetic functions, grouped by levels $k$. Fix some $k$ and some family $\varepsilon_*=(\varepsilon_\sigma)_{\...
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0answers
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 ...
3
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0answers
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 : $ \...
0
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1answer
92 views

Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and $D(N^2)$ is the deficiency of $N^2$?

Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and $$D(N^2)=2N^2 - \sigma(N^2)$$ is the deficiency of $N^2$? I checked OEIS sequence A033879 and have so far been able to get hold of ...
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1answer
278 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

I posted this question on MSE two days ago, but did not receive any responses. I have cross-posted it on MO, hoping it gets more attention here and that it is appropriate for this site. A positive ...
3
votes
1answer
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=\...
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1answer
374 views

There at least 4 divisors of $n-1$ which do not divide $\phi(n)$ if $n$ is a composite of the form $6k+1$

If $n$ is composite then $\phi(n) < n-1$ (Euler's totient function) hence there must be one or more divisors of $n-1$ which do not divide $\phi(n)$. For lack of a better terminology, let us call ...
27
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2answers
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 ...
1
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1answer
233 views

Is it possible to have an even superperfect number and an odd superperfect number whose product is an almost perfect number?

A number $n \in \mathbb{N}$ is said to be superperfect if $$\sigma(\sigma(n)) = 2n.$$ A number $m \in \mathbb{N}$ is said to be almost perfect if $$\sigma(m) = 2m - 1.$$ Here is my question: Is ...
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0answers
232 views

On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.) Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...
15
votes
3answers
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 ...
0
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1answer
114 views

What is the relative size of the radical of an ABC-triple relative to the number of primes up to its largest element?

Write $\bf N$ for the set of natural numbers, and $P$ for the set of primes. For $x$ in $\bf N$ let $p(x)$ be the product of the primes dividing $x$ (that is, the "radical" of $x$). Also write $\#(x)$ ...
0
votes
2answers
193 views

Is there an example of integers ($x,p, q ,y$ ) which satisfies the below conditions in this claim? [closed]

Edit 01:In order to look divisibility among power divisor function where i would like to know if there a such integer $n>1 $ with y coprime to $x$ then we have: :$\sigma_y(n)\bmod \sigma_x(n)=0$, ...
1
vote
1answer
191 views

For which $x$ and $y$ does $\sigma_x(n) $ divide $\sigma_y(n)$ for all $n$?

I would like to know more about divisibility among power-divisor functions. Put $\sigma_k(n) = \sum_{d \mid n} d^k$ for all positive integers $k$ and $n$. My question here is : for which positive ...
4
votes
1answer
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{...
0
votes
0answers
136 views

Sum of reciprocals of primitive sequences with distinct prime factors

In a previous mathoverflow question here a construction of a primitive sequence $1<a_1<\cdots<a_k\leq n$ formed by including all the integers in $[1,n]$ with exactly $k$ prime divisors (...