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.
236 questions
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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 $(a^n+1,b^n+1)=1$ ?
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Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?
This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered.
Let $\...
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Sum of divisors and unitary divisors as the eigenvalue and the spectral norm of some addition matrix?
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this set to a ring by observing that each divisor $d$ has
$$0 \le v_p(d) \le v_p(n)$$
Hence we can add two divisors $d,e$ by ...
user6671
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Lower bound on a Truncated Divisor Sum
Let $d(n)$ be the number of divisors function, i.e., $d(n)=\sum_{k\mid n} 1$ of the positive integer $n$.
I am interested in estimating, the following sum
$$
A(a,x)=\sum_{n\leq x} \min[ d(n), M]^a
$$
...
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Restricted divisor summatory function
I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where
$$a_n:=1+\sum_{0\leq j<n}D_{2j+1}(n-j)$$
and $D_{m}(n)$ is the number of divisors $d>1$ of $n$ ...
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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:
$$S(x)=\sum_{i=1}...
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The kronecker symbol and factorization of $n=\frac{B^N-1}{B-1}$
Let $n=\frac{B^N-1}{B-1}$. Assume $n$ is congruent to 3 modulo 4.
We have the following:
If $N$ is 1 modulo 4, then $N$ is quadratic residue modulo $n$
and $-N$ is quadratic non-residue. The square ...
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1
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Generalized Erdős multiplication table problem
Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
What is the cardinality of the range?
At $k =2$ ...
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"Oddity" of Fibonacci-Catalan numbers
As a follow up to my previous two MO questions, here and here, let's consider the below inquiry.
Define the Fibonacci-Catalan numbers by $FC_n=\frac1{F_{n+1}}\binom{2n}n_F$ where $F_0=0, F_1=1, F_0!=...
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How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free?
I am interested in obtaining an upper bound for $\prod_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that
$$
\prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...
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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}(...
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Is the asymptotic density of positive integers $n$ satisfying $\gcd(n, \sigma(n^2))=\gcd(n^2, \sigma(n^2))$ equal to zero?
(This post is an offshoot of this MSE question.)
Let $\sigma(x)$ denote the sum of divisors of $x$. (https://oeis.org/A000203)
QUESTION
Is the asymptotic density of positive integers $n$ satisfying $...
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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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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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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\...
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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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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 ...
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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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Generalized Thomas Ordowski conjecture at OEIS sequence A002326
OEIS is the online encyclopedia of integer sequences, Here is the link to the sequence $A002326$: https://oeis.org/A002326
For $n\geq 0$, the $n$th term in the sequence is defined as: $a(n)$ equals ...
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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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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:
$$D_k(x) = \sum_{n \leq x} \tau_k(n) = xP_k(\log x) + O(x ^{1 - \frac{1}{k-1}}(\...
- 151
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Solutions of the equation $\psi(\sigma(n))=2n$, where $\sigma(n)$ is the sum of divisors function and $\psi(n)$ the Dedekind psi function
For integers $m\geq 1$ let $\sigma(m)$ the sum of divisors function $\sum_{1\leq d\mid m}d$ and let $\psi(m)$ the Dedekind psi function (as reference I add the Wikipedia Dedekind psi function), then ...
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Touchard / van der Pol's identity for the sum of divisors and an elliptic curve for perfect numbers
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$, satisifies the following recurrence relation ($n>1$):
$$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
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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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On $\mathsf{LCM}$ of a set of integers
For integers $a,b$ define $$\mathcal R(a,b)=\{q\in\mathbb Z\cap[1,\min(a,b)]: a\equiv b\bmod q\}$$ and $\mathsf{LCM}(\mathcal R(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal R(a,b)$.
How ...
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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 ...
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g.c.d. and Euler's totient function
There is this really nice paper by J.P.Serre on the congruence subgroup property for $SL_2$ for $S$-arithmetic groups (https://www.jstor.org/stable/1970630). If one looks at the proof of Proposition 3 ...
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Are there an infinite number of integers $n$ such that $n, n+1$, and $n+2$ have the same number of divisors?
Is the set $S:=\{n\in\mathbb{N} \mid \text{$n$, $n+1$ and $n+2$ have the same number of divisors}\}$ infinite?
Example: $33\in S$.
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1
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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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Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$?
This question, comes out of a question in MSE and I hope it is ok to ask it here:
Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$?
...
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On the $\mathsf{LCM}$ of a set of integers defined by moduli of powers
For integers $a,b,t$ define $$\mathcal R_t(a,b)=\{q\in\mathbb Z\cap[1,\min(a^t,b^t)]: a^t\equiv b^t\bmod q\}$$ and $\mathsf{LCM}(\mathcal R_t(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal ...
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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 ...
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If $n$ is a multiperfect number, then necessarily does one of its prime factors $p$ satisfy $p \parallel n$?
My question is as in the title:
If $n$ is a multiperfect number, then necessarily does one of its prime factors $p$ satisfy $p \parallel n$?
I quote from an answer by Varun Vejalla to a closely ...
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Does iterating a certain function related to the sums of divisors eventually always result in a prime value?
Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...
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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
$$
S_1(x)=\sum_{n\leq x} d(n)...
- 10.4k
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Equations involving arithmetic functions of primorials
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors, $\varphi(n)$ the Euler's totient function and we denote the primorial $\prod_{k=1}^n p_k$ as $N_n$, where $p_k$ denotes the $k$-th prime ...
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Counting multiples in short intervals
Has anyone seen a problem like this in the literature? There are likely more generalized versions in sieve theory, which I am willing to tackle, but I would prefer a more elementary approach if ...
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Are there pairs of consecutive integers with the same sum of factors?
Background/Motivation
I was planning to explain Ruth-Aaron pairs to my son, but it took me a few moments to remember the definition. Along the way, I thought of the mis-definition, a pair of ...
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Smooth integers with lower bound on $\omega(n)$
Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.
Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
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Attempt of exploit the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$ in the context of even perfect numbers, and a related conjecture
It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $\varphi(n)$ denotes ...
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1
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186
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Around a characterization for even perfect numbers, similar than Euclides-Euler theorem, in terms of totatives
In this post we denote the sum of divisors function as $$\sigma(n)=\sum_{1\leq d\mid n}d,$$ then an even perfect number is a positive integer $n\equiv 0\text{ mod }2$ for which $\sigma(n)=2n.$ As ...
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Near Pochhammer symbols: the equation $(n)_m-(k)_l=2$ for integers greater than or equal to two
In this post I consider the following equation involving Pochhammer symbols,
$$(n)_m-(k)_l=2\tag{1}$$
for positive integers $n\geq 2$ and $k\geq 2$, and positive integers $m\geq 2$ and $l\geq 2$.
...
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3
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On the number of consecutive divisors of an integer
Define for $n \in \mathbb{N}$ the function $$\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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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 ...
1
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1
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213
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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 ...
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4
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Sum of the sum-of-divisors function
I was looking at the abstract of a paper 1 which claims that [2] and [3] prove
$$
\sum_{n\le x}\sigma(n)-\frac{\pi^2}{12}x^2=\Omega(x\log\log x).
$$
But I cannot find the above—or indeed, ...
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1
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284
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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
$$\ll\frac{\text{exp}\left(C\frac{\log(N)}{\log\log(N)}\right)}{A^...
- 23
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0
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Least number of factors $\sigma(p^e)$ of representation of $\sigma(N)$ to get the least multiple of $\operatorname{rad}(N)$, for odd perfect numbers
I've cross-posted this from the post of Mathematics Stack Exchange that I've asked (Apr, 2nd 2020) with title On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\...
0
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0
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759
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On sets of coprime integers in intervals
Briefly,
Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval?
The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
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On a variant of Brocard's problem using the definition of Pochhammer symbols
I've considered the following variant of Brocard's problem $$\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 ...