# 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.

136
questions

**-1**

votes

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49 views

### Erdos multiplication table problem avoiding integers with too many distinct prime factors

Consider multiplication operation $$f(x_1,x_2)=x_1x_2$$ where $x_i\in\{1,\dots, n_i\}\backslash T_i$ with $n_1, n_2\in\{1,\dots,\infty\} $ where $T_i$ is set of positive integers less than $n_i$ which ...

**-1**

votes

**0**answers

45 views

### Distribution of table entries in Erdos multiplication table problem

Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
At $k =2$ with $n_1=n_2$ this is the standard Erdos ...

**0**

votes

**1**answer

128 views

### Generalized Erdős multiplication table problem

Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
What is the cardinality of the range?
At $k =2$ ...

**1**

vote

**1**answer

121 views

### Bounds for two arithmetic functions, when one assumes that $n$ are odd perfect numbers

For an integer $n>1$ in this post we denote the Dedekind psi function as $\psi(n)=n\prod_{\substack{p\mid n\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$ and the product of distinct primes dividing ...

**4**

votes

**0**answers

152 views

### A generalization of Feit–Thompson conjecture, for square-free integers

I asked the following question with my account that I have for these sites Mathematics Stack Exchange and MathOverflow. The bounty that I offered in MSE expired without answers. The post that I refer ...

**2**

votes

**0**answers

155 views

### The multiplicative constant in the estimate for $S_a(x)=\sum_{n\leq x} d(n)^a$

Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
S_a(x)=\sum_{n\leq x} d(n)^a.
$$
For $a=1,$ the following is well known
$$
S_1(x)=\sum_{n\leq x} d(n)...

**6**

votes

**0**answers

152 views

### Smooth integers with lower bound on $\omega(n)$

Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.
Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...

**5**

votes

**1**answer

148 views

### The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem

Let $n\geq 1$ be an integer. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are ...

**1**

vote

**0**answers

209 views

### On variants of the abc conjecture in terms of Lehmer means

In this post we denote the Lehmer mean of a tuple $\text{x}$ of positive real numbers as $$L_p(\text{x})={\sum_{k=1}^nx_k^p\over\sum_{k=1}^nx_k^{p-1}},$$
see the reference Wikipedia Lehmer mean.
The ...

**8**

votes

**0**answers

579 views

### Divisor problem: find the fallacy!

The following approach to the divisor problem (that is, the problem of estimating $D(x) = \sum_{n\leq x} d(n)$, where $d(n)$ is the number of divisors of $n$; more precisely, we are meant to bound the ...

**25**

votes

**3**answers

3k views

### Expressing the Riemann Zeta function in terms of GCD and LCM

Is the following claim true: Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,
$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\...

**1**

vote

**1**answer

130 views

### Counting multiples in short intervals

Has anyone seen a problem like this in the literature? There are likely more generalized versions in sieve theory, which I am willing to tackle, but I would prefer a more elementary approach if ...

**17**

votes

**2**answers

1k views

### Is every prime the largest prime factor in some prime gap?

Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other ...

**0**

votes

**1**answer

98 views

### Diophantine equations that involve cubes and the volume of square frustums

This week I wondered about diophantine problems that involve the volume of certain cubes and frustums, see the Wikipedia Frustum. I wondered if each one of these problems have infinitely many ...

**1**

vote

**1**answer

149 views

### Solutions of the equation $\psi(\sigma(n))=2n$, where $\sigma(n)$ is the sum of divisors function and $\psi(n)$ the Dedekind psi function

For integers $m\geq 1$ let $\sigma(m)$ the sum of divisors function $\sum_{1\leq d\mid m}d$ and let $\psi(m)$ the Dedekind psi function (as reference I add the Wikipedia Dedekind psi function), then ...

**3**

votes

**1**answer

196 views

### How to estimate the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?

Let $\tau(n)$ be the number of positive divisors of $n\in \mathbb{N}$.
Is it possible to get some good estimate for the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?
I know that the sum is $\mathcal O(x^2)$...

**1**

vote

**1**answer

128 views

### Around a characterization for even perfect numbers, similar than Euclides-Euler theorem, in terms of totatives

In this post we denote the sum of divisors function as $$\sigma(n)=\sum_{1\leq d\mid n}d,$$ then an even perfect number is a positive integer $n\equiv 0\text{ mod }2$ for which $\sigma(n)=2n.$ As ...

**0**

votes

**0**answers

66 views

### Partial sums involving Gregory coefficients that cannot be an integer

For integers $n\geq 1$ let $G_n$ be the Gregory coefficients or reciprocal logarithmic numbers, see the Wikipedia [Gregory coefficients.]
(https://en.wikipedia.org/wiki/Gregory_coefficients)
$${z\...

**0**

votes

**1**answer

110 views

### On a variant of Brocard's problem using the definition of Pochhammer symbols

I've considered the following variant of Brocard's problem $$\frac{(2n-1)!}{(n-1)!}+1=m^2\tag{1}$$
for integers $n\geq 1$ and integers $m\geq 1$. I was inspired from the fact that the evaluation of ...

**1**

vote

**1**answer

102 views

### A problem inspired in the definition of tau numbers and a divisibility relationship related to powers of two

It is (I assume that this easy fact is well-known) obvious that an integer $n>1$ is a power of two $n=2^{\alpha}$, where $\alpha\geq 1$ is integer, if an only if $n$ satisfies the divisibility ...

**2**

votes

**0**answers

76 views

### Catalan numbers, Pochhammer symbols, Stirling numbers of the second kind, and sums of aliquot parts

For integers $N\geq 1$ we define $$s(N)=\sigma(N)-N$$ the aliquot sum function, where $\sigma(N)=\sum_{1\leq d|N}d$ is the sum of divisors function.
Here $(x)_n$ is the Pochhammer symbol and ${a\...

**7**

votes

**1**answer

356 views

### Do the following binary vectors span $\mathbb{R}^n$?

Defining the binary vectors
Let an ordered triple of natural numbers $(r, d, n)$ such that $0 \leq r < d \leq n$ be given.
Consider the binary vector $v_{(r,d,n)} \in \mathbb{R}^n$ such that for ...

**0**

votes

**0**answers

56 views

### On the number of solutions of the equation involving Pochhammer symbols $(n)_a\cdot(n)_b=(n)_c$, for integers greater than or equal to $2$

As paticular case of the equation involving Pochhammer symbols $$(n)_a\cdot(m)_b=(k)_c,$$
where the variables are positive integers, I've consider the case $n=m=k$ of previous equation, that is
$$(n)...

**1**

vote

**0**answers

60 views

### Two conjectures inspired from an equation involving the sum of divisors and the Euler's totient function due to Iannucci

In this post I add two equations involving the sum of divisors $\sigma(n)$ and the Euler's totient function, denoted in this post as $\varphi(n)$, and after I ask about a conjecture involving these. ...

**2**

votes

**1**answer

163 views

### Equations involving arithmetic functions of primorials

Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors, $\varphi(n)$ the Euler's totient function and we denote the primorial $\prod_{k=1}^n p_k$ as $N_n$, where $p_k$ denotes the $k$-th prime ...

**4**

votes

**0**answers

126 views

### What is known about the mode of the number of divisors $\le x$?

Let $d(x)$ be the divisor function. Let $M(x)$ ($x$ a positive integer) be the most frequent value of $d(y)$ for $1 \le y \le x$. Is an asymptotic known for $M(x)$, and failing that, can $M(x)$ at ...

**0**

votes

**1**answer

124 views

### Sum of divisors of Stirling numbers of the second kind

In this post we denote the Stirling number of the second kind as ${n\brace k}$, I add as reference the article Stirling numbers of the second kind from the encyclopedia Wikipedia. And we denote the ...

**4**

votes

**0**answers

139 views

### Near Pochhammer symbols: the equation $(n)_m-(k)_l=2$ for integers greater than or equal to two

In this post I consider the following equation involving Pochhammer symbols,
$$(n)_m-(k)_l=2\tag{1}$$
for positive integers $n\geq 2$ and $k\geq 2$, and positive integers $m\geq 2$ and $l\geq 2$.
...

**0**

votes

**0**answers

39 views

### Approximation of $\sum_{\substack{n\geq 1\\n\text{ is abundant}}}\frac{\sigma(n)}{n^3}$, where $\sigma(n)$ denotes the sum of divisors function

Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors function then, from the theory of Dirichlet series, it is well-known the value of $$\sum_{n=1}^\infty\frac{\sigma(n)}{n^3},$$
in terms of ...

**4**

votes

**2**answers

206 views

### Estimating $\sum_{n\leq x: n \in A} d(n)^a$ from below for large sets $A\subset \{1,2,\ldots,x\}.$

I apologise for the long-windedness of this question.
Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
S_a(x)=\sum_{n\leq x} d(n)^a.
$$
For $a=1,$ ...

**1**

vote

**0**answers

44 views

### On the equation involving Stirling numbers of the second kind ${n\brace a}{m\brace b}={k\brace c}$, and its solutions satisfying certain requirements

In this post we denote the Stirling numbers of the second kind as ${r\brace s}$ and we consider the proposal to ask if the equation of the title has infinitely many solutions $${n\brace a}{m\brace b}={...

**1**

vote

**0**answers

51 views

### On characterizations for Mersenne primes involving the sum of divisor function

In this post we denote the sum of positive divisors function of an integer $n\geq 1$ as $$\sigma(n)=\sum_{1\leq d\mid n}d.$$
Then a prime of the form $2^p-1$ is called a Mersenne prime. These are ...

**2**

votes

**0**answers

45 views

### Is it possible to deduce statements for odd perfect numbers from the convolution sums involving divisor functions or other arithmetic functions?

Dividing and using some identities of [1] I've deduced the following facts, see also the remarks below. After these introductory paragraphs, to motivate our question, I am asking if we can deduce some ...

**2**

votes

**1**answer

175 views

### On a problem that equates $\frac{\text{prime}-1}{\operatorname{rad}(\text{prime}-1)}$ with the sequence of primorials

We denote for integers $m>1$ the product of the distinct prime numbers dividing $m$ as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(...

**11**

votes

**3**answers

562 views

### Number of matrices with bounded products of rows and columns

Fix an integer $d \geq 2$ and for every real number $x$ let $M_d(x)$ be number of $d \times d$ matrices $(a_{ij})$ satisfying: every $a_{ij}$ is a positive integer, the product of every row does not ...

**1**

vote

**0**answers

36 views

### Equations involving quasiperfect numbers: a first search of odd solutions for this type of equations or well succinct reasonings about these

In this post we study the following equations that involve quasiperfect numbers, denoted as $x$, that are integers such that the sum of all its positive divisors is equals to $2x+1$, and certain ...

**1**

vote

**0**answers

19 views

### Asymptotic size for the number of terms not exceeding $n$ in the class $r$ for a classification of the type Erdös-Selfridge for square-free integers

It is possible to define a classification similar than the Erdös-Selfridge classification of primes for different sequences. Please ee [1], section A18 and the references cited in this book. Because ...

**1**

vote

**0**answers

192 views

### On the number of solutions of $\gcd\left({2n\brace n},105\right)=1$, over integers $n\geq 1$

In this post we denote the Stirling numbers of the second kind as ${n\brace k}$. I present a variant of the problem showed in the penultimate paragraph of section B33 of [1] (see also the cited ...

**2**

votes

**0**answers

47 views

### On $\sum_{\substack{1\leq d\mid n\\d<f(n)}}d$ and odd perfect numbers, for $f(n)$ the greatest prime factor or $\operatorname{rad}(n)$, respectively

First, in this paragraph we remember the definitions/notations for two number theoretic functions, for an integer $m>1$, we denote its greatest prime factor as $\operatorname{gpf}(m)$, and the ...

**3**

votes

**0**answers

117 views

### An attempt to get a variant of Agoh–Giuga conjecture

The idea of this post is an attempt to explore a variant of the so-called Agoh–Giuga conjecture. In past days, and today, I tried to think about variants of this conjecture exploring congruences about ...

**3**

votes

**1**answer

260 views

### Greatest common divisor of $(a^n+1,b^n+1)$

Let $(a,b)$ be a pair of coprime positive integers with $a$ being even. Are these conditions sufficient to prove that there exist infinitely many positive integers $n,$ such that $(a^n+1,b^n+1)=1$ ?

**1**

vote

**0**answers

100 views

### Size of a set defined by divisor function

After some computations, I guessed the following conjecture.
How can I prove or disprove it? thanks!
Let
$$
A(k)=\#\left\{\left(t,\frac{k+t+a}{4t-1}\right):1\leq t\leq k,\ 1\leq a\leq k+t,\ a\mid(k+...

**2**

votes

**0**answers

97 views

### Queries on distribution of prime divisors by magnitude?

Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors and we know probability of square free integers is $\frac{6}{\pi^2}$.
What is the probability distribution of ...

**3**

votes

**0**answers

114 views

### Magnitude and distribution of largest prime factor?

Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors.
What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number?
What ...

**3**

votes

**1**answer

144 views

### $\sum_{i=1}^x\sum_{j=1}^xf(i\cdot j)$ Double Summing a (Not Completely) Multiplicative Function

Let $f(n)$ be a multiplicative function that is not completely multiplicative, i.e $f(m)\cdot f(n)= f(m\cdot n)$ only if $gcd(m,n)=1$. Let $S(x)$ be the double sum over $f$, that is:
$$S(x)=\sum_{i=1}...

**1**

vote

**0**answers

101 views

### Structure of set with large pairwise gcd's

Let $\mathcal{M}$ be a set of $M$ distinct positive integers, all of size roughly $N$. Assume that the pairwise gcd of elements of $\mathcal{M}$ is large for all pairs. For illustration, let's take $M ...

**1**

vote

**1**answer

128 views

### Runs of consecutive numbers that are not relatively prime to their digital sum

It is well known that there can be at most 20 consecutive integers (in base 10) that are divisible by their digital sum, so called Harshad or Niven numbers.
How long can a run of consecutive ...

**1**

vote

**0**answers

51 views

### On divisibility conditions implying local coprimality conditions

This question is inspired by Bernardo Recaman's question Strings of consecutive integers divisible by 1, 2, 3, ..., N on intervals of $n$ integers being divisible by the integers $1$ through $n$. The ...

**5**

votes

**1**answer

236 views

### “Oddity” of Fibonacci-Catalan numbers

As a follow up to my previous two MO questions, here and here, let's consider the below inquiry.
Define the Fibonacci-Catalan numbers by $FC_n=\frac1{F_{n+1}}\binom{2n}n_F$ where $F_0=0, F_1=1, F_0!=...

**1**

vote

**1**answer

104 views

### Question about arithmetic binomial coefficient

i have a question about the following assertion:
let $n,j,u $ positive integer satisfying
$ n \geq 5,$ $ 1\leq j \leq n-1$,$ \; n+1 \leq u \leq n+j$
let $ d[n]:=\operatorname{lcm}[1,2,..,n]$ ...