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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If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$?

My present question is as is in the title: If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$? It is known that $m^2 - p^k$ is ...
Jose Arnaldo Bebita Dris's user avatar
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If $p^k m^2$ is an odd perfect number with special prime $p$, is it possible to have $p = k$?

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. My question is as is in the title: If $p^k m^2$ is an odd perfect number with special prime $p$, is it ...
Jose Arnaldo Bebita Dris's user avatar
6 votes
2 answers
634 views

Number of divisors which are at most $n$

I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by $$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$ the number of divisors of $x$ which are at most $n$. Question 6 of ...
TheBestMagician's user avatar
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1 answer
398 views

On a GCD approach to odd perfect numbers

Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Let $\sigma(z)$ denote the classical sum of divisors of the positive ...
Jose Arnaldo Bebita Dris's user avatar
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1 answer
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What can be said about $\gcd(N/q^{\alpha},\sigma(N/q^{\alpha}))$ where $N$ is an odd perfect number and $q^{\alpha} \parallel N$?

What can be said about the quantity $$\gcd(N/q^{\alpha},\sigma(N/q^{\alpha}))$$ where $N$ is an odd perfect number and $q^{\alpha} \parallel N$? In particular, can one prove that it is always greater ...
Jose Arnaldo Bebita Dris's user avatar
4 votes
1 answer
195 views

Representation of a number as a product of $\sqrt{n^2 + 1} + n$

Question. Do there exist two multisets $A, B$ consisting of positive integer numbers such that $|A|$ and $|B|$ have different parity and $$ \prod_{n\in A}(n + \sqrt{n^2 + 1}) = \prod_{m\in B}(m + \...
Pavel Gubkin's user avatar
1 vote
0 answers
100 views

Mysterious recursion for the A005225

Let $a(n)$ be A005225 i.e. number of permutations of length $n$ with equal cycles. Here $$ a(n)=n!\sum\limits_{d|n}\frac{1}{d!(\frac{n}{d})^d} $$ Let $$ R(n,q,z)=(q+1)R(n-1,q+1,z)+\sum\limits_{j=0}^{q}...
Notamathematician's user avatar
7 votes
1 answer
189 views

Are there infinite numbers of the form $\sigma_1(n)=\sigma_1(m)=p$, or is there only one?

I put forward a hypothesis in number theory, it is as follows.$ \sigma_1(n)=\sigma_1(m)=p$, where $\sigma_1$ is the divisor sum function, $n,m\in \mathbb N$, and $p$ is prime. I recently noticed and ...
Arsen Vardanyan's user avatar
2 votes
0 answers
130 views

Exponential sum of $k$-fold divisor function

Can anyone point me to a reference for the main term when approximating the exponential sum of the 3-fold divisor function? Specifically I want the main term in $$\sum _{n\leq x}d_3(n)e\left (an/q\...
tomos's user avatar
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2 votes
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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 ...
joaopa's user avatar
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7 votes
2 answers
795 views

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 ...
Adithya Chakravarthy's user avatar
9 votes
1 answer
502 views

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 ...
Adithya Chakravarthy's user avatar
1 vote
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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, ...
Jose Arnaldo Bebita Dris's user avatar
4 votes
0 answers
76 views

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 ...
user142929's user avatar
3 votes
1 answer
942 views

On the OEIS sequence A327265

The OEIS sequence https://oeis.org/A327265 starts: $$1, 2, 5, 11, 19, 31, 51, 89, 123, 151, 179, 181, 180, 365, 634, 657, 656, 655.$$ $\mathrm{A327265}(n)$ is the smallest $k$ such that $\mathrm{...
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Given that $H = \frac{n^2}{\sigma(q^k)/2} = G \times J^2$, where $q^k n^2$ is an odd perfect number, then what is the value of $\gcd(G, J)$?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
Jose Arnaldo Bebita Dris's user avatar
3 votes
0 answers
67 views

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 $[x,x+\log^2(x)]$ ...
G. Melfi's user avatar
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4 votes
2 answers
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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 $\mathbb F_p$ ...
Feldmann Denis's user avatar
4 votes
0 answers
104 views

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 $a(k,n)$...
Fabius Wiesner's user avatar
1 vote
2 answers
193 views

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 ...
Mikhail Bondarko's user avatar
4 votes
1 answer
235 views

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 ...
JoshuaZ's user avatar
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1 vote
0 answers
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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 ...
JoshuaZ's user avatar
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8 votes
1 answer
278 views

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 ...
Zach Hunter's user avatar
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4 votes
4 answers
438 views

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) ...
user142929's user avatar
3 votes
1 answer
116 views

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 ...
Alexey Staroletov's user avatar
-9 votes
1 answer
497 views

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 ...
user142929's user avatar
1 vote
1 answer
142 views

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 ...
user142929's user avatar
0 votes
1 answer
194 views

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) - ...
Jose Arnaldo Bebita Dris's user avatar
3 votes
1 answer
199 views

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 $x^2-2$ ...
user142929's user avatar
5 votes
1 answer
186 views

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 $\sum_{b\in B} \frac{n}{b}=O(n)$?
Chao Xu's user avatar
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2 votes
1 answer
333 views

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 ...
user142929's user avatar
1 vote
0 answers
144 views

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 $m>1$ ...
user142929's user avatar
4 votes
1 answer
248 views

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 ...
user142929's user avatar
2 votes
1 answer
296 views

Analogue of Fermat's little theorem for Bernoulli numbers

Is the following analogue of Fermat's Little Theorem for Bernoulli numbers true? Let $D_{2n}$ be the denominator of $\frac{B_{2n}}{4n}$ where $B_n$ is the $n$-th Bernoulli number. If $\gcd(a, D_{2n}) ...
Nilotpal Kanti Sinha's user avatar
2 votes
0 answers
107 views

How to compute/estimate the least $k$ such that there exist $n$ consecutive integers each having a prime factor $\le k$?

Let $a_n$ be the least integer $k$ such that there exist $n$ consecutive integers each with a prime factor $\le k$. For example, $a_{13} \le 11$ because the 13 consecutive integers $114,115,\ldots,126$...
tuna's user avatar
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3 votes
0 answers
176 views

Does this Theorem 2 from Dandapat et al. imply that $\gcd(\sigma(p^k),\sigma(a^2)) > 1$?

Write the odd perfect number $m=p^k a^2$ as a product of primes $$m = p^k {p_1}^{2a_1} \cdots {p_v}^{2a_v}.$$ (Note that it is known that $v \geq 9$ by work of Nielsen.) Let $N(m)$ be the number of ...
Jose Arnaldo Bebita Dris's user avatar
3 votes
2 answers
233 views

Help with R. Ryan's "A simpler dense proof regarding the abundancy Index."

I'm reading Richard Ryan's article "A simpler dense proof regarding the abundancy index" and got stuck in his proof for Theorem 2. The Theorem is stated as follows: Suppose we have a ...
jvkloc's user avatar
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2 votes
0 answers
84 views

Estermann's argument for the binary additive divisor problem

In the paper https://eudml.org/doc/149759 an estimate for the binary additive divisor problem is given with a power saving. I don't get the main bit of the argument - I'm obviously missing something. ...
tomos's user avatar
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3 votes
0 answers
175 views

The binary additive divisor problem in arithmetic progressions

I find quite a few results about the binary additive divisor problem, that is evaluating \[ \sum _{n\leq x}d(n)d(n+h)\] for certain ranges of $h$. Are there any known results about the same count ...
tomos's user avatar
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3 votes
0 answers
79 views

Estimating from below positive moments of "clipped" divisor function on subsets of $\{1,2,\ldots,x\}$ with positive density

The question here about estimating positive moments of the divisor function on sets of nonzero density $A\subset \{1,2,\ldots,x\}$ was answered giving $$ S_a(x):=\sum_{n \in A} d(n)^a \geq |A|(\ln x)^{...
kodlu's user avatar
  • 10.1k
5 votes
1 answer
255 views

Analogue of the second Hardy-Littlewood conjecture for numbers of divisors?

Let $f(n)$ denote the proposition "There exists some $k>1$ such that $$ \sum_{m=k}^{k+n-1}\tau(m) < \sum_{m=1}^n\tau(m) $$ where $\tau(m)$ is the number of the divisors of $m$." (This ...
Charles's user avatar
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1 vote
0 answers
293 views

About inequalities that involve the sum of divisors, the Euler's totient and the aliquot part $\sigma(n)-n$

In this post, for integers $n\geq 1$, I denote the sum of divisors $\sum_{1\leq d\mid n}d$ as $\sigma(n)$ and the Euler's totient function as $\varphi(n)$. It's easy to check* that if we assume that $...
user142929's user avatar
2 votes
2 answers
474 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II

(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.) The topic of odd perfect ...
Jose Arnaldo Bebita Dris's user avatar
0 votes
0 answers
106 views

On improving the upper bound $I(m^2) \leq \frac{2p}{p+1}$, if $p^k m^2$ is an odd perfect number with special prime $p$

(Preamble: This question is an offshoot of this answer to an MSE question with the same title.) Denote the classical sum of the divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$ and the ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
0 answers
723 views

Does $n \mid \sigma(n^2)$, if $q^k n^2$ is an odd perfect number?

Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. It is known that $$\gcd(\sigma(q^k),\sigma(n^2))=\frac{\bigg(\gcd(n,\sigma(n^2))\bigg)^2}{\gcd(n^2,\sigma(n^2))...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
316 views

On odd perfect numbers and a GCD - Part III

Let $m = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. It is known that $$\gcd(\sigma(q^k),\sigma(n^2)) = \frac{(\gcd(n,\sigma(n^...
Jose Arnaldo Bebita Dris's user avatar
0 votes
0 answers
137 views

A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem

I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
user142929's user avatar
3 votes
1 answer
297 views

Behavior of biggest prime divisor of $n$ as $n$ grows large

Let $P\subseteq \mathbb{N}$ be the set of primes, and for any integer $n>1$ let $L(n) = \max\{p \in P: p \mid n\}$ be the largest prime divisor of $n$. Moreover, for $n \in \mathbb{N}$ with $n>1$...
Dominic van der Zypen's user avatar
14 votes
2 answers
860 views

How many divisors of $n$ are below $n^{1/3}$?

I am trying to bound a function that includes $\sum\limits_{\substack{d < n^{1/3} \\ d \mid n}} 1$. Is there an upper bound known for this sum, either in general or in terms of $\sum\limits_{\...
Nico Tripeny's user avatar
0 votes
1 answer
168 views

Proportionality constant in Montgomery-Vaughan Theorem 7.20

In Multiplicative Number Theory - Vol. I by Montgomery and Vaughan the following result is proved. Theorem 7.20 Let $A(x,r)$ denote the number of $n\leq x$ such that $\Omega(n)\leq r \log \log x,$ and ...
kodlu's user avatar
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