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.
225
questions
5
votes
0
answers
157
views
A modern reference for the Piltz divisor problem
apparently, the Dirichlet hyperbola method is no longer up-to-date, and instead Voronoi's identity is used in order to establish good bounds on the Dirichlet divisor problem.
The same applies to the ...
- 547
1
vote
0
answers
100
views
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
$$
A(a,x)=\sum_{n\leq x} \min[ d(n), M]^a
$$
...
- 10.1k
6
votes
1
answer
331
views
Factors of polynomials of bounded height
Let $f(x)=a_nx^n+\cdots+a_0 \in \mathbb{Z}[x]$ be an integer polynomial in one variable. Recall that the height $H(f):=\textrm{max}\,|a_n|$ is the largest coefficient. Consider the set of polynomials ...
- 1,493
2
votes
1
answer
336
views
On odd perfect numbers $q^k n^2$ satisfying $n^2 - q^k = 2^r t$
Let $N = q^k n^2$ be an odd perfect number with special prime $q$, satisfying
$$n^2 - q^k = 2^r t$$
where $r \geq 2$ and $\gcd(2,t)=1$.
We could prove that:
(1) $2^r t > 2n$. (We can modestly ...
- 1,008
1
vote
2
answers
366
views
Finding all proper divisors of $a_3z^3 +a_2z^2 +a_1z+1$ of the form $xz+1$
Let $n=a_3z^3+a_2z^2+a_1z+1$ where $a_1<z, \ a_2<z, \ 1 \le a_3<z, z>1$ are non negative integers. To obtain proper divisors of $n$ of the form $xz+1$, one may perform trial divisions $xz+...
- 319
11
votes
2
answers
892
views
Have any proposals been advanced for the analytic continuation of the divisor function?
While I was working on the evaluation of a certain series, the following limit came up:
\begin{align} \lim_{n \to 1} \frac{d(n)-1}{n(n-1)} &= \lim_{n \to 1} \frac{d'(n)}{2n-1} \\
&= d'(1) .\...
- 4,575
6
votes
0
answers
217
views
Growth rate of signed sum of $N \sigma_0(n)-\sigma_1(n)$
Let $\sigma_k(n)=\sum_{d|n} d^k,$ for a positive integer $n$ and $k\geq 0$. A lot is known about the averages for the functions $\sigma_k(n),$ such as the estimates
$$
\sum_{n\leq x} \sigma_0(n)=x \...
- 10.1k
5
votes
1
answer
340
views
Can the Lagarias inequality be written as a "kernel inequality"?
The Lagarias inequality, which is equivalent to the Riemann hypothesis, is:
$$\sigma(n) \le H_n + \exp(H_n) \log(H_n) =:L(n)$$
for all natural numbers $n$, where $\sigma=$ sum of divisors, $H_n=n$-th ...
- 2,462
1
vote
2
answers
383
views
Improving the lower bound $I(n^2) > \frac{2(q-1)}{q}$ when $q^k n^2$ is an odd perfect number
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Define the abundancy index
$$I(x)=\frac{\sigma(x)}{x}$$
where $\sigma(x)$ ...
- 1,008
1
vote
1
answer
231
views
Moments of number of interval restricted divisors
I have previously asked the question A truncated divisor function sum
where the sum
$$
S_f(x)=\sum_{n\leq x} \min\{f(x),d(n)\}\quad (1)
$$
was of interest, and it was answered satisfactorily.
Here, I ...
- 10.1k
3
votes
0
answers
165
views
The kronecker symbol and factorization of $n=\frac{B^N-1}{B-1}$
Let $n=\frac{B^N-1}{B-1}$. Assume $n$ is congruent to 3 modulo 4.
We have the following:
If $N$ is 1 modulo 4, then $N$ is quadratic residue modulo $n$
and $-N$ is quadratic non-residue. The square ...
- 24.2k
3
votes
1
answer
429
views
Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?
Robin's inequality
$$\sigma_1(n)<e^\gamma n\log\log n$$
at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma_1(n)=\sum_{d|n}d$ is sum of divisors ...
- 13.7k
2
votes
0
answers
84
views
On consecutive superabundant numbers
Define $\sigma(n)=\sum_{d\mid n} d$. A number $n>1$ is said to be superabundant (SA) if it is an integer and $\frac{\sigma(n)}{n}>\frac{\sigma(s)}{s}$ for every positive integer $s<n$. Let $n$...
- 21
2
votes
0
answers
339
views
On odd perfect numbers and a GCD - Part II
(Note: A detailed version of this question was posted in MSE last April 15, 2020. It has not received any responses there as of yet. I have therefore cross-posted it here, hoping that it is ...
- 1,008
8
votes
0
answers
265
views
Restricted divisor summatory function
I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where
$$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$ ...
- 177
2
votes
2
answers
627
views
On the nearest-square function and the quantity $m^2 - p^k$ where $p^k m^2$ is an odd perfect number
This question has been cross-posted from this MSE question and is an offshoot of this other MSE question.
(Note that MSE user mathlove has posted an answer in MSE, which I could not completely ...
- 1,008
2
votes
0
answers
90
views
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$?
...
- 2,462
1
vote
0
answers
185
views
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 ...
5
votes
1
answer
284
views
Is the factorization of $a_m-a_n$ affected by the fact that $\Sigma \frac{1}{a_k}<+\infty$?
I would like to ask the following.
Let $(a_n)$ be a sequence of natural numbers such that
$\sum_{k=1}^{\infty}\frac{1}{a_k}$ converges. Is it true that for
infinitely many $m$, there is a $n<m$ ...
- 3,826
2
votes
1
answer
615
views
$\frac{\sigma(n)}{n} < e \ln \ln (n)$ is true?
In Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213 (pdf)
we find the following result:
If the Riemann hypothesis is true ...
- 53
7
votes
0
answers
164
views
Occurrence of binary words in divisibility patterns
Given an integer $n \geq 1$, let $d_n : \mathbb{N}_{\geq 1} \to \{0,1\}$ be the coloring of the positive integers defined by $d_n(x) = 1$ if $x \mathbin{|} n$ and $d_n(x) = 0$ otherwise.
In other ...
- 740
7
votes
2
answers
565
views
Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?
This question is related to the last question about van der Pol's identity for the sum of divisors.
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
- 2,462
9
votes
0
answers
673
views
Van der Pol's identity for the sum of divisors and a quartic polynomial equation for odd perfect numbers
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies 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)\...
- 213
5
votes
0
answers
164
views
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 ($n>1$):
$$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
- 213
6
votes
0
answers
501
views
Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?
This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered.
Let $\...
- 1,008
4
votes
1
answer
198
views
Sum of divisors and unitary divisors as the eigenvalue and the spectral norm of some addition matrix?
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this set to a ring by observing that each divisor $d$ has
$$0 \le v_p(d) \le v_p(n)$$
Hence we can add two divisors $d,e$ by ...
user6671
2
votes
1
answer
166
views
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 $...
- 1,008
7
votes
1
answer
362
views
If $n = 18k+5$ is composite, there are at least 9 divisors of $\phi(n)$ which do not divide $n-1$
If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example? The conjecture has ...
- 4,638
2
votes
0
answers
111
views
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 ...
- 1,008
6
votes
0
answers
277
views
Is there a positive odd $n$ such that $\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$?
Let $\sigma(n)$ denote the sum of the divisors of $n$. (https://oeis.org/A000203)
It is relatively easy to find numbers $n$ such that $f(g(n)) = g(f(n))$ where $f(n) = \sigma(n)$ and $g(n) = \sigma(n) ...
- 681
6
votes
0
answers
195
views
On nontotient Fibonacci numbers
This question is related to sequence of numbers $t$ such that $F_{6t}$ is a nontotient where $F_n$ represents the sequence of Fibonacci numbers for $n\geq 0$.
The online encyclopedia Wikipedia has the ...
- 681
2
votes
0
answers
74
views
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
votes
0
answers
152
views
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\}$$ and $\mathsf{LCM}(\mathcal R_t(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal ...
- 1,816
0
votes
1
answer
142
views
On $\mathsf{LCM}$ of a set of integers
For integers $a,b$ define $$\mathcal R(a,b)=\{q\in\mathbb Z\cap[1,\min(a,b)]: a\equiv b\bmod q\}$$ and $\mathsf{LCM}(\mathcal R(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal R(a,b)$.
How ...
- 1,816
3
votes
1
answer
364
views
Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes
I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and ...
0
votes
1
answer
246
views
Generalized Erdős multiplication table problem
Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
What is the cardinality of the range?
At $k =2$ ...
- 1,816
1
vote
1
answer
189
views
Bounds for two arithmetic functions, when one assumes that $n$ are odd perfect numbers
For an integer $n>1$ in this post we denote the Dedekind psi function as $\psi(n)=n\prod_{\substack{p\mid n\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$ and the product of distinct primes dividing ...
8
votes
0
answers
339
views
A generalization of Feit–Thompson conjecture, for square-free integers
I asked the following question with my account that I have for these sites Mathematics Stack Exchange and MathOverflow. The bounty that I offered in MSE expired without answers. The post that I refer ...
2
votes
0
answers
191
views
The multiplicative constant in the estimate for $S_a(x)=\sum_{n\leq x} d(n)^a$
Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
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)...
- 10.1k
6
votes
0
answers
197
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 ...
- 1,816
7
votes
1
answer
221
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 ...
0
votes
0
answers
260
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
631
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 ...
- 19.4k
28
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,
$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\...
- 4,638
1
vote
1
answer
196
views
Counting multiples in short intervals
Has anyone seen a problem like this in the literature? There are likely more generalized versions in sieve theory, which I am willing to tackle, but I would prefer a more elementary approach if ...
- 13k
20
votes
2
answers
2k
views
Is every prime the largest prime factor in some prime gap?
Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other ...
- 4,638
0
votes
1
answer
136
views
Diophantine equations that involve cubes and the volume of square frustums
This week I wondered about diophantine problems that involve the volume of certain cubes and frustums, see the Wikipedia Frustum. I wondered if each one of these problems have infinitely many ...
0
votes
1
answer
283
views
Solutions of the equation $\psi(\sigma(n))=2n$, where $\sigma(n)$ is the sum of divisors function and $\psi(n)$ the Dedekind psi function
For integers $m\geq 1$ let $\sigma(m)$ the sum of divisors function $\sum_{1\leq d\mid m}d$ and let $\psi(m)$ the Dedekind psi function (as reference I add the Wikipedia Dedekind psi function), then ...
3
votes
1
answer
289
views
How to estimate the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?
Let $\tau(n)$ be the number of positive divisors of $n\in \mathbb{N}$.
Is it possible to get some good estimate for the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?
I know that the sum is $\mathcal O(x^2)$...
- 3,826
1
vote
1
answer
179
views
Around a characterization for even perfect numbers, similar than Euclides-Euler theorem, in terms of totatives
In this post we denote the sum of divisors function as $$\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 ...