Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

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34 views

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 ...
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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 ...
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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)...
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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 ...
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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)...
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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 $y$ ...
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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 ...
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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}{\...
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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 ...
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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 ...
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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 ...
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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 $<1$, ...
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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 ...
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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 $x$ ...
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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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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 ...
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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 ...
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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 ...
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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 \...
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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 $\pi(x)$ ...
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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 $n$). ...
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1answer
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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 ...
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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 ...
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2answers
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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) ...
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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
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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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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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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 $f(N)$ ...
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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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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 = \...
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1answer
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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 $1$. ...
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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 (...
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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 $n$....
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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 $\sigma(x)$ ...
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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,...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 $...
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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 $...
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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 ...
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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 $$...
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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,...
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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 "...
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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 $$...
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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
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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_{...
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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 $\...

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