# 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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### 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 ...
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### 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 ...
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### 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 $k =2$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathcal O(x^2)$...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $l\geq 2$. ...
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### 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 ...
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### 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 $a=1,$ ...
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### 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 ...
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### 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 ...
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### $\sum_{i=1}^x\sum_{j=1}^xf(i\cdot j)$ 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}...
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 $M ... 1answer 128 views ### Runs of consecutive numbers that are not relatively prime to their digital sum It is well known that there can be at most 20 consecutive integers (in base 10) that are divisible by their digital sum, so called Harshad or Niven numbers. How long can a run of consecutive ... 0answers 51 views ### On divisibility conditions implying local coprimality conditions This question is inspired by Bernardo Recaman's question Strings of consecutive integers divisible by 1, 2, 3, ..., N on intervals of$n$integers being divisible by the integers$1$through$n$. The ... 1answer 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!=...
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 $d[n]:=\operatorname{lcm}[1,2,..,n]$ ...