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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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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?...
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1answer
122 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....
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0answers
112 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(...
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156 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 ...
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1answer
148 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 ...
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51 views

Best method to compute sum of divisors of bounded evaluation of a bivariate quadratic?

Given a bivariate quadratic polynomial $g(u,v)\in\mathbb Z[u,v]$ and an integer $n$. How fast can we compute $\sum_{i=-\ell}^{\ell}\sum_{j=-\ell'}^{\ell'}\sigma_0(g(i,j))\bmod2$ where $\sigma_0$ is ...
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1answer
265 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 ...
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150 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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208 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 ...
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79 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 $$\...
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2answers
189 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 ...
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1answer
104 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.
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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 ...
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3answers
286 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 ...
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88 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\...
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1answer
607 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 ...
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1answer
312 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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1answer
176 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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1answer
215 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 ...
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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 ...
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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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2answers
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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^{\...
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446 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 ...
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0answers
197 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 : $ \...
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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 ...
49
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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 ...
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1answer
361 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 ...
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572 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 ...
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1answer
371 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 ...
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1answer
206 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 ...
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1answer
222 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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230 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 ...
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2answers
802 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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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 $\#(x)$ ...
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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, ...
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2answers
191 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: :$\sigma_y(n)\bmod \sigma_x(n)=0$, ...
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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 ...
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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{...
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131 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 (...
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1answer
447 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 ...
3
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1answer
290 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
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,...
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103 views

Consecutive integers divisible by consecutive small numbers

Given $n$, what is the largest set of consecutive integers in $[n,2n]$ can we have so that each integer is divisible by a distinct element from $[\log n,2\log n]$ (no partiular order)? So apriori I am ...
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2answers
622 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:...
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1answer
299 views

Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime? [closed]

I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^p$$ such that $m=k{n}^{p-1}$ with $m,n>0$ and $p$ is an odd prime? Note: $\sigma(\frac{m}{{n}^{p-1}})$ is the sum of ...
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1answer
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(\...
11
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2answers
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. $\...
12
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3answers
654 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 ...
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1answer
276 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 ...
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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 ...