Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
696
questions
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How to write this sum $\sum_{m=0}^\infty (2m+1)^{-2k-1} \sum_{r=1}^\infty (-1)^{r-1} e^{-2(2m+1)r a} $ as a sum over single index?
So I want to write the sum $$\sum_{m=0}^\infty (2m+1)^{-2k-1} \sum_{r=1}^\infty (-1)^{r-1} e^{-2(2m+1)r a} $$ where $a>0$ and $k\in \mathbb{N}$, as a sum over single index which probably uses odd ...
0
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1answer
49 views
Resources for asymptotic analysis in analytic number theory?
Currently, I'm looking for some good resources on the methods of asymptotic analysis that can be applied in analytic number theory. So far, I've found books on asymptotic analysis (e.g. the book by de ...
1
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1answer
95 views
On an inequality involving the abundancy index of the Eulerian component $p^k$ of an odd perfect number $p^k m^2$
The topic of odd perfect numbers likely needs no introduction.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the abundancy index of $x$ by $I(x)=\sigma(x)...
4
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0answers
55 views
Lower bound on $\frac{f(σ(n))}{f(n)}$ where $n = \prod\limits_i p_i^{e_i}, f(n) = \prod\limits_i \frac{p_i+1}{p_i}$ and $σ$ is the divisor function.
If $n$ has the prime factorization $\prod\limits_i p_i^{e_i}$ for distinct primes $p_i$ and exponents $e_i > 0,$ the sum of divisors is $\sigma(n) = \prod\limits_i \frac{p_i^{e_i+1}-1}{p_i-1}.$ I'm ...
2
votes
1answer
98 views
If $p^k m^2$ is an odd perfect number with special prime $p$, then which is larger: $D(p^k)$ or $D(m^2)$?
Denote the classical sum of divisors of the positive integer $x$ to be $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. Finally, denote the deficiency of $x$ by $D(x)...
3
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202 views
On odd perfect numbers $p^k m^2$ with special prime $p$, satisfying $m^2 - p^k = 2^r t$
In what follows, we will denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. A positive integer $y$ ...
1
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1answer
59 views
Partial summation of $d(n)/(n-1) $
In the answers to this question, it is established that $$\sum_{n\leq x}\frac{d(n)}{n}=\frac{1}{2}(\log(x))^{2}+2\gamma\log (x)+\gamma^{2}-2\gamma_{1}+O\left(x^{-1/2}\right).$$ A related result can be ...
2
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1answer
69 views
Average order of $\phi(n)$
Theorem 3.7 of the book Analytic Number Theory by Apostol states:
$$\sum_{n\le x} \phi(n)= \frac{3}{\pi^2} x^2 + O(x\log x)$$
and then it claims : Hence the average order of $\phi(n)$ is $\frac{3n}{\...
2
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0answers
61 views
Average Number of Small Divisors
I'm working on a pet project of mine and I've come across a seemingly simple problem that I can neither solve nor find any reference to in the literature. The problem is this: Given $x$ sufficiently ...
1
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2answers
106 views
If $p^k m^2$ is an odd perfect number, then is there a constant $D$ such that $\frac{\sigma(m^2)}{p^k} > \frac{m^2 - p^k}{D}$?
(Note: This question is an offshoot of this closely related one.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
The topic of odd perfect numbers likely ...
2
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1answer
63 views
Follow-up to question 3121498, asked in February 2019
Let $n = p^k m^2$ be an odd perfect number given in Eulerian form (i.e. $p$ is the special/Euler prime satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$). Denote the classical sum of ...
2
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1answer
48 views
For sum of divisors of 4n+1 and 4n+3 and is there a known inequality with n?
I was trying to understand the relationship between $S(n)$ and $S(4n+1)$ and between $S(n)$ and $S(4n+3)$.
It seems that up to $10^5$, the ratios $S(4n+1)/S(n)$ and $S(4n+3)/S(n)$ are always $<1$, ...
3
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0answers
32 views
How to find smallest sum of denominators so that sum of fractions is less than or equal to 1, given the numerators? [closed]
Given $n$ positive integer numbers $a_1, a_2, \dots, a_n$, how do I find $n$ positive integer numbers $b_1, b_2, \dots, b_n$, so that
$$\frac{a_1}{b_1} + \frac{a_2}{b_2} + \dots + \frac{a_n}{b_n} \leq ...
1
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1answer
46 views
On the quantity $I(q^k) + I(n^2)$ considered as functions of $q$ and $k$, when $q^k n^2$ is an odd perfect number with special prime $q$
Hereinafter, let $N = q^k n^2$ be an odd perfect number with special/Euler prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Denote the abundancy index of the positive integer $x$ ...
2
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3answers
103 views
Applying a criterion on deficient numbers to the proper factors of an odd perfect number
Hereinafter, let $I(x)=\sigma(x)/x$ denote the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$. Denote the deficiency of $x$ by
$$D(...
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1answer
57 views
A proof (?) for $k = 1 \implies q \neq 5$, if $q^k n^2$ is an odd perfect number with special prime $q$
Let $N = q^k n^2$ be an odd perfect number given in Eulerian form (i.e. $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$).
Inspired by mathlove's answer to ...
1
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1answer
131 views
On bounds for the quantity $n^2 / D(n^2)$ when $q^k n^2$ is an odd perfect number with special prime $q$ and $k > 1$
Let $N = q^k n^2$ be an odd perfect number given in Eulerian form (i.e. $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$).
In this post, I would like to ...
0
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0answers
49 views
On improving $D(q^k)/q^k > (q-2)/(q-1)$, if $D(x)=2x-\sigma(x)$ and $q$ is a prime number
It is known that, for $q$ prime and $k$ a positive integer,
$$I(q^k) = \frac{\sigma(q^k)}{q^k} = \frac{1 + q + \ldots + q^k}{q^k} = \frac{q^{k+1} - 1}{q^k (q - 1)},$$
where $\sigma(x)=\sigma_1(x)$ is ...
1
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1answer
62 views
Is there an odd $x$ such that $2x^2 \equiv 0 \pmod {\sigma(x)}$ and $\sigma(x^2) \equiv 0 \pmod {\sigma(x) - 1}$?
CONTEXT
This question is a result of considerations stemming from this closely related MO question.
INITIAL QUESTION
My question is as is in the title:
Is there an odd $x$ such that $$2x^2 \equiv 0 \...
1
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0answers
45 views
Help Understanding A Statement by D.H.J. POLYMATH
Question: D.H.J. POLYMATH wrote in the paper Deterministic Methods To Find Primes the following statement:
"$\ldots$ the key observation is that the parity of the prime counting function $\pi(x)$ ...
3
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1answer
166 views
Abundant products of iterations of Euler's totient function
Let $a_0(n) = n$ and $a_{i+1}(n) = \varphi(a_i(n))$ for $i\geq 0$, where $\varphi(n)$ is Euler's totient function (the number of positive integers less than or equal to $n$ and coprime with $n$). ...
1
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1answer
41 views
If $k + 1$ is prime and $(k + 1) \mid (q - 1)$, then $\sigma(q^k)$ is divisible by $k + 1$, but not by $(k + 1)^2$ (unless $k+1=2$).
I tried Googling for the keywords "the theory of odd perfect numbers" and one of the search results that came up was this document, titled ON THE DIVISORS OF THE SUM OF A GEOMETRICAL SERIES ...
1
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1answer
38 views
Square of number of divisors of n equals n
Find $n \in \mathbb{N}$ such that $n = \tau(n)^2$ ($\tau(n)$ being the number of positive divisors of $n$).
I tried some values for $n$, it seems that besides $n = 1$ and $n = 9$ there's no other ...
1
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2answers
47 views
The average order of $\sigma(n)$.
I'm reading through Apostol's wonderful book "Introduction to Analytic Number Theory" but became confused when reading the statement of theorem 3.4. It states that
$$
\sum_{n\leq x}\sigma(n) ...
3
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0answers
499 views
On $\frac{D(n^2)}{s(q)} \geq \frac{2n^2}{\sigma(q)} \geq \frac{\sigma(n^2)}{q} \geq \frac{2s(n^2)}{D(q)}$ where $q^k n^2$ 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 $\gcd(q,n)=1$. Descartes, Frenicle, and subsequently Sorli conjectured that $k=1$ always ...
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1answer
76 views
On the inequality $\frac{\sigma(n^2)}{q^k} < \frac{n^2 - q^k}{C}$ where $C>1$ and $q^k n^2$ is an odd perfect number - Part II
(Note: This is a continuation of this earlier question.)
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. (In particular, ...
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0answers
85 views
Is $n^2\mid \sigma(n)^{\sigma(n)}-1$ possible for $n>1$?
Inspired by Martin Hopf I used another number theoretical function for this question.
If $\sigma(n)$ is the divisor-sum function ($1$ and $n$ are included, $\sigma(1)=1$) , can we have $$n^ 2\mid \...
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0answers
15 views
How can the following summation be written in terms of $N$?
Suppose that $N=\sum_{j=1}^{Q}k_j,$ where the $0\leq k_j$ are integers. Then, what is $\sum_{j=1}^{Q}jk_j$ in terms of $N$ and possibly $Q$. Meaning, if
$$f=\sum_{j=1}^{Q}jk_j,$$
what is either $f(N)$ ...
2
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0answers
24 views
Is there a similar relation to Robin's Inequality for the number of divisors ( $\sigma_0(n) $), instead of the sum of divisors (( $\sigma_1(n) $)
Robin's Inequality states:
$\sigma(n)\lt e^\gamma n\log\log n $
for any n>5040.
Is there an equivalent for $\sigma_0(n) $?
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1answer
60 views
Prove that exist finitely many positive integers $n$ satisfying $\tau (n)=a$ and $n|\phi (n)+\sigma (n)$
Given a fixed positive integer $a\geq 9$. Prove there exist finitely many positive integers $n$, satisfying:
$\tau (n)=a$
$n|\varphi (n)+\sigma (n)$
My ideal is if i write the factorization $ n = \...
-1
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1answer
91 views
Number theory olympiad problem [closed]
This is a problem I have been tackling recently, but I am unsure how to address it.
A positive integer $n$ is good if there exists a set of divisors of $n$ whose members sum to $n$ and include $1$. ...
2
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1answer
45 views
About $\lim\dfrac{\sigma(n)}{n}$, where $\sigma(n)$ is the sum of divisor of $n$
Denote $p\equiv 1 \mod 3$ is a prime number and $n$ is any number that divisible by $p$. Denote $\sigma(n)$ is the sum of divisor of $n$.
We know that $\dfrac{\sigma (n)}{n}>1$ and
$$\dfrac{\sigma (...
5
votes
2answers
99 views
Prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$
Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$.
($\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$....
4
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2answers
362 views
Does the following lower bound improve on $I(q^k) + I(n^2) > 3 - \frac{q-2}{q(q-1)}$, where $q^k n^2$ 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 $\sigma(x)$ ...
4
votes
1answer
55 views
coefficient $[q^n]\sum_{m\ge1}\frac{q^{\alpha m}}{1-q^{\beta m}}$ where $0<\alpha<\beta$
Problem:
I am looking for a finite-sum expression for the coefficient $c_n=c_n(\alpha,\beta)$, where
$$C(\alpha,\beta;q)=\sum_{m\ge1}\frac{q^{\alpha m}}{1-q^{\beta m}}=\sum_{n\ge1}c_n(\alpha,\beta)q^n,...
2
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1answer
191 views
What numerical lower bound on the index of an odd perfect number is implied by the results in F.-J. Chen and Y.-G. Chen's 2014 paper?
In what follows, we let $\sigma=\sigma_1$ denote the classical sum-of-divisors function.
Suppose that $N$ is an odd perfect number and $q^\alpha$ is a prime power with $q^\alpha \parallel N$.
(Euler ...
1
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1answer
247 views
Revisiting questions 3888565 and 3894925
The topic of odd perfect numbers likely needs no introduction.
I would like to revisit these two questions:
Is it possible to improve on the bound $D(q^k) < \varphi(q^k)$ if $k>1$?
On the ...
2
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1answer
60 views
Property of a divisor function, $\lim_{n\to\infty} \frac{\rho(n, n)}{n^2}=1-{\pi^2\over12}$ where $\rho(n, k):=\sum_{i=1}^k{(n\mod i)}$
Let $$\rho(n, k):=\sum_{i=1}^k{(n\mod i)}$$ where $(n \mod i)$ is a remainder of $n$ divided by $k$. Also let $\sigma(n)$ be the sum of all factors of $n$, or $$\sigma(n):=\sum_{d|n} d$$ i.e. a ...
1
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1answer
46 views
When is the number of Divisors of a Number equivalent to one of its Factors?
My math teacher asked me this problem for homework and I am unsure how to solve it.
Which numbers contain a number of factors equivalent to the value of one of their divisors?
I found that 8 works, ...
1
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0answers
53 views
If $q^k n^2$ is an odd perfect number with special prime $q$, is $\sigma(q^k)$ coprime to $\sigma(n^2)$?
Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Here is my:
QUESTION
If $q^k n^2$ is an odd perfect number with special prime ...
1
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1answer
87 views
Question on sums of products of divisors (cont.)
In my previous Question on sums of products of divisors, I asked the following question:
Does it exist some positive integer $n$ such that a sum of $k$ products of its divisors greater than $1$ equal $...
1
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1answer
244 views
Question on sums of products of divisors
Let us consider some positive integer $n$ with divisors (other than $1$) $d_1, d_2, ..., d_n$. Lately, it came to me the following question:
Does it exist some positive integer $n$ such that a sum of $...
0
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0answers
29 views
Divisor sum of totient function squared
I was wondering if it's possible to express the following sum as a function of $n$:
$$
\sum_{d|n} [\phi(d)]^2
$$
similarly to the known relation:
$$
\sum_{d|n} \phi(d) = n
$$
where $\phi$ is the ...
2
votes
1answer
101 views
Is every multiperfect number also pseudoperfect?
It seems like something that should be pretty obvious but I don't quite get why would it be true. For example, in the case of 2-fold perfect numbers, or simply perfect numbers, it is evident because
$$...
-1
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1answer
65 views
Question about a set of distinct odd integers and their least common multiple
Let $S=\{x_1,x_2,...,x_n\}$ be some set of distinct odd integers greater than 1, such that $\forall x_i\in S$ it holds that $x_i<\sqrt{\text{lcm} (x_1,x_2,...x_n)}$ and $x_i\mid \text{lcm} (x_1,...
0
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1answer
170 views
Why does an odd perfect number seemingly “violate” basic inequality rules?
Let $p^k m^2$ be an odd perfect number (OPN) with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
My question is as is in the title:
Why does an OPN seemingly "...
1
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1answer
95 views
On a curious equation regarding odd perfect numbers
Let $p^k m^2$ be an odd perfect number (OPN) with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
Since $\gcd(p^k, \sigma(p^k))=1$, then we essentially get the equation
$$...
2
votes
3answers
221 views
If $p^k m^2$ is an odd perfect number, then what is the optimal constant $C$ such that $\frac{\sigma(m^2)}{p^k} < \frac{m^2 - p^k}{C}$?
The topic of odd perfect numbers likely needs no introduction.
The question is as is in the title:
If $p^k m^2$ is an odd perfect number with special prime $p$, then what is the optimal constant $C$ ...
2
votes
1answer
42 views
Does this inequality hold for all odd $n$: $\sigma_{n-1}\sigma_{n+3} > \sigma_{n}\sigma_{n+2}$?
Let be $\sigma_n = \sum\limits_{d|n}d$ and let $n$ be an odd number, then the following inequality holds numerically up to $n = 10^7$.
$$
\begin{align}
\sigma_{n-1}\sigma_{n+3} > \sigma_{n}\sigma_{...
1
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1answer
60 views
Estimate of $\prod_{d\mid n}(d+1)$.
I would like to ask if there is any cool estimate of $\prod_{d\mid n}(d+1)$.
I know that $\prod_{d\mid n}d=n^{\tau(n)/2}$ ($\tau(n)$ is the number of divisors of $n$) so we have the trivial estimate $\...
