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
7
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1
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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 ...
0
votes
0
answers
264
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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 ...
8
votes
0
answers
643
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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 ...
- 20.2k
28
votes
3
answers
3k
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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)}{\...
- 5,470
0
votes
1
answer
224
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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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20
votes
2
answers
2k
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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 ...
- 5,470
0
votes
1
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140
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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 ...
0
votes
1
answer
296
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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 ...
3
votes
1
answer
309
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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)$...
- 3,876
1
vote
1
answer
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,$$ then an even perfect number is a positive integer $n\equiv 0\text{ mod }2$ for which $\sigma(n)=2n.$ As ...
0
votes
0
answers
89
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Partial sums involving Gregory coefficients that cannot be an integer
For integers $n\geq 1$ let $G_n$ be the Gregory coefficients or reciprocal logarithmic numbers, see the Wikipedia [Gregory coefficients.]
(https://en.wikipedia.org/wiki/Gregory_coefficients)
$${z\...
0
votes
1
answer
201
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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 ...
0
votes
1
answer
212
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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 ...
3
votes
0
answers
285
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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\...
7
votes
1
answer
502
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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 ...
- 123
0
votes
0
answers
64
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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
$$(n)...
2
votes
0
answers
98
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Two conjectures inspired from an equation involving the sum of divisors and the Euler's totient function due to Iannucci
In this post I add two equations involving the sum of divisors $\sigma(n)$ and the Euler's totient function, denoted in this post as $\varphi(n)$, and after I ask about a conjecture involving these. ...
2
votes
1
answer
231
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Equations involving arithmetic functions of primorials
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors, $\varphi(n)$ the Euler's totient function and we denote the primorial $\prod_{k=1}^n p_k$ as $N_n$, where $p_k$ denotes the $k$-th prime ...
5
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0
answers
235
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What is known about the mode of the number of divisors $\le x$?
Let $d(x)$ be the divisor function. Let $M(x)$ ($x$ a positive integer) be the most frequent value of $d(y)$ for $1 \le y \le x$. Is an asymptotic known for $M(x)$, and failing that, can $M(x)$ at ...
- 251
1
vote
1
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258
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Sum of divisors of Stirling numbers of the second kind
In this post we denote the Stirling number of the second kind as ${n\brace k}$, I add as reference the article Stirling numbers of the second kind from the encyclopedia Wikipedia. And we denote the ...
4
votes
0
answers
185
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Near Pochhammer symbols: the equation $(n)_m-(k)_l=2$ for integers greater than or equal to two
In this post I consider the following equation involving Pochhammer symbols,
$$(n)_m-(k)_l=2\tag{1}$$
for positive integers $n\geq 2$ and $k\geq 2$, and positive integers $m\geq 2$ and $l\geq 2$.
...
0
votes
0
answers
47
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Approximation of $\sum_{\substack{n\geq 1\\n\text{ is abundant}}}\frac{\sigma(n)}{n^3}$, where $\sigma(n)$ denotes the sum of divisors function
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors function then, from the theory of Dirichlet series, it is well-known the value of $$\sum_{n=1}^\infty\frac{\sigma(n)}{n^3},$$
in terms of ...
5
votes
2
answers
403
views
Estimating $\sum_{n\leq x: n \in A} d(n)^a$ from below for large sets $A\subset \{1,2,\ldots,x\}.$
I apologise for the long-windedness of this question.
Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
S_a(x)=\sum_{n\leq x} d(n)^a.
$$
For $a=1,$ ...
- 10.4k
1
vote
0
answers
63
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On the equation involving Stirling numbers of the second kind ${n\brace a}{m\brace b}={k\brace c}$, and its solutions satisfying certain requirements
In this post we denote the Stirling numbers of the second kind as ${r\brace s}$ and we consider the proposal to ask if the equation of the title has infinitely many solutions $${n\brace a}{m\brace b}={...
1
vote
0
answers
65
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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
68
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
280
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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}(...
11
votes
3
answers
605
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 ...
- 213
1
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0
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56
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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
28
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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
207
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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
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0
answers
57
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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
299
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
330
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
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0
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222
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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 ...
1
vote
0
answers
101
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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
$$
A(k)=\#\left\{\left(t,\frac{k+t+a}{4t-1}\right):1\leq t\leq k,\ 1\leq a\leq k+t,\ a\mid(k+...
- 841
2
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0
answers
112
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Queries on distribution of prime divisors by magnitude?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors and we know probability of square free integers is $\frac{6}{\pi^2}$.
What is the probability distribution of ...
- 13.9k
3
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0
answers
280
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Magnitude and distribution of largest prime factor?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors.
What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number?
What is ...
- 13.9k
4
votes
1
answer
375
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$\sum_{i=1}^x\sum_{j=1}^xf(i\cdot j)$ Double Summing a (Not Completely) Multiplicative Function
Let $f(n)$ be a multiplicative function that is not completely multiplicative, i.e $f(m)\cdot f(n)= f(m\cdot n)$ only if $gcd(m,n)=1$. Let $S(x)$ be the double sum over $f$, that is:
$$S(x)=\sum_{i=1}...
- 311
1
vote
0
answers
134
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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 ...
- 2,227
1
vote
1
answer
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 ...
- 3,717
1
vote
0
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57
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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 ...
- 13k
5
votes
1
answer
318
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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!=...
- 43.2k
1
vote
1
answer
111
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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]$ ...
- 417
6
votes
1
answer
258
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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
$$
\prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...
- 3,625
4
votes
1
answer
646
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Piltz Divisor Problem
Let $\tau_k(n)$ count the number of ways of representing $n$ as the product of $k$ natural numbers. It is known that:
$$D_k(x) = \sum_{n \leq x} \tau_k(n) = xP_k(\log x) + O(x ^{1 - \frac{1}{k-1}}(\...
- 151
2
votes
1
answer
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
$$\ll\frac{\text{exp}\left(C\frac{\log(N)}{\log\log(N)}\right)}{A^...
- 23
0
votes
0
answers
759
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On sets of coprime integers in intervals
Briefly,
Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval?
The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
- 13k
10
votes
2
answers
759
views
Find all $m$ such $2^m+1\mid5^m-1$
The problem comes from a problem I encountered when I wrote the article
Find all positive integer $m$ such
$$2^{m}+1\mid5^m-1$$
it seem there no solution. I think it might be necessary to use ...
- 4,280
1
vote
0
answers
290
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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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