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Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

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Sum of divisors of $n$ less than $k$

It is easy to know the sum of divisors of $n$ just by calculating the prime factorization of $n$. Is it possible to calculate the sum of divisors of $n$ that is less than $k$ ($k<n$) without ...
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About The Sum of Positive Divisors of $n$

The question says: Find the smallest positive integer $n$ so that $\sigma(x)=n$ has no solution, exactly two solutions, exactly three solutions. I could not come up with a good way to solve this ...
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A bridge between the sum of the divisors and the Totient function

Let's consider: $$\tau(x,a,b)=\sum_{1 \le d \le x \\ (d,x)=d^a \\} d^b$$ Where $(q,r)$ denotes the gcd of $q$ and $r$. I think this could be interesting thing to look at because it's somehow a ...
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1answer
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What proportion of the positive integers satisfies $\gcd(n^2,\sigma(n^2))>\sigma(n)$, where $\sigma$ is the sum-of-divisors function?

The title says it all. What proportion of the positive integers satisfies $\gcd(n^2,\sigma(n^2))>\sigma(n)$, where $\sigma$ is the sum-of-divisors function? I tried using Sage Cell Server to ...
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A proof of $\sum\limits_{d|n} \sigma(d) = n \sum\limits_{d|n} {\tau(d) \over d}$

I'm trying to proof the following statement: Let $n \in \mathbb{Z}$ and the $\sum$ are on the divisors $d$ of $n$. Show that $$\sum\limits_{d|n} \sigma(d) = n \sum\limits_{d|n} {\tau(d) \over d}.$$ ...
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1answer
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A question regarding a paper of Ochem and Rao about the radical of an odd perfect number

Let $\operatorname{rad}(n)$ denote the radical or square-free part of the positive integer $n$, that is, $$\operatorname{rad}(n) = \prod_{p \mid n}{p}$$ where $p$ runs over primes. In the paper ...
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A functional ecuation using divizors

It’s a functional equation. We have a function f defined on pozitive integers (greater than 0) with values on real numbers. Also, for any pozitive (and non zero) integer n. It asks to find function f. ...
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Asymptotic behavior $\sum_{n=1}^x\phi_k(n)$, a variant of Euler's Totient function

Let $$\phi_k(x)=\sum_{1\le n \le x \\(n,x)=1} n^k$$ What's the asymptotic behavior of $$\sum_{n=1}^x\phi_k(n)?$$ According to the wikipedia $\sum^x_{n=1} \phi_0 (n) \approx \frac{3}{\pi^2}x^2 $. It ...
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1answer
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The sum of all divisors of $N$ is a 2 potency iff $N$ is a product of different Mersenne primes

As far as I have controlled: $\sigma(a)=2^n$, for some $n\in\mathbb N \iff $ $a$ is a product of different Mersenne primes. The $\Leftarrow$-part is an immediate consequence of that $\sigma$ is ...
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An anomaly regarding the sum of all divisors that is square-free

Let $q(n)$ be the number of integers $m<n$ such that $m$ is square-free. Let $p(n)$ be the number of integers $m<n$ such that the sum of the prime factors of $m$ is square-free. And let $s(n)$ ...
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Divisor sum simplification

How does this : $$D(n)=\displaystyle\sum\limits_{i=1}^n i \left\lfloor\frac{n}{i}\right\rfloor$$ become $$D(n)=\displaystyle\sum\limits_{i=1}^{n/(u+1)} i \left\lfloor\frac{n}{i}\right\rfloor + \sum_{d=...
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Degree of a canonical divisor on a compact Riemann surface

I'm reading Jürgen Jost's "Compact Riemann Surfaces" Springer textbook 3rd ed (a very good read!). Jost defines the divisor of a meromorphic differential $\eta$ on a compact Riemann surface by \...
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On the quantity $\sigma(\frac{n^2 \sigma(n^2)}{D(n^2)})$ when $q n^2$ is an odd perfect number with special prime $q$

Denote the sum of the divisors of $x \in \mathbb{N}$ by $\sigma(x)$. Also, denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$. If $m$ is odd and $\sigma(m)=2m$, then $m$ s called an odd perfect ...
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For what $k\in \mathbb{C}$ do we have $\lim_{x\to\infty}\frac{1}{x^{k+1}}\sum_{n=1}^x \sigma_k(n)=\frac{\zeta(k+1)}{k+1}$?

Can the following claim be extended for some complex $k$. Perhaps: all $k$ with real part greater than or equal to $1$? Do the arguments below fall apart for complex $k$ for some reason? Claim. ...
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202 views

On $\phi(n) \sigma(n)$ being a square.

I can't understand underlined statements: The green one: For p an odd number, both p-1 and p+1 are even so all prime factors in $\prod (p-1)(p+1)$ must be belong to the set B, so the prime factors ...
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The asymptotic behavior of $\sum_{n=1}^x\sigma_a(n)\sigma_b(n)$

Question What is the asymptotic behavior of $$\sum_{n=1}^x\sigma_a(n)\sigma_b(n)$$ Where $\sigma_k=\sum_{d|n}d^k$ More generally I am curious if we can get bounds on $$\sum_{n=1}^x\prod_{i}\sigma_{...
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If $x$ and $y$ are solitary numbers satisfying $\gcd(x,y)=1$, under what conditions does it follow that $xy$ is also solitary?

Let $\sigma(z)$ denote the sum of divisors of $z \in \mathbb{N}$. Denote the abundancy index of $z$ by $I(z) = \sigma(z)/z$. If the equation $I(z)=I(a)$ has the lone solution $z=a$, then $a$ is said ...
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Is comparing real and complex values within Robin's Inequality legal? And how would we?

I have a problem where I need to compare real and complex numbers. I see here and here that there are different ways to go about interpreting the sizes of complex numbers, but in my context I want to ...
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Asymptotic formula for $\sum_{an+b\le x}\sigma(an+b)$

In this post $\sigma(n):=\sum_{d|x}{d}$ which is called the divisor function or sometimes the sum-of-the-divisors function. Is there an asymptotic formula for $$f(a,b,x):=\sum_{an+b\le x}\sigma(an+b)...
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1answer
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Can this bound for the abundancy index of $n$ be improved, given that $q^k n^2$ is an odd perfect number with $k=1$?

In what follows, set $I(x)=\sigma(x)/x$ to be the abundancy index of $x \in \mathbb{N}$, where $\sigma(x)$ is the sum of divisors of $x$. If $I(y)=2$ and $y$ is odd, then $y$ is called an odd perfect ...
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Could a Fermat prime divide an odd perfect number?

This question is a follow-up to this earlier post. A positive integer $N$ is said to be perfect if $\sigma(N)=2N$, where $\sigma(x)$ is the sum of divisors of $x \in \mathbb{N}$. If $M$ is odd and $\...
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Does mathlove's answer imply that $D(n^2) \mid n^2$?

In what follows, set $\sigma(x)$ to be the sum of divisors of $x \in \mathbb{N}$, and let $$D(x) = 2x - \sigma(x)$$ be the deficiency of $x$, and let $$s(x) = \sigma(x) - x$$ be the sum of the aliquot ...
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1answer
126 views

Asymptotic formula for $\sum_{n\leq x}\sigma(n)$ knowing $\sum_{n\leq x}\frac{\sigma(n)}{n}$

Let $\sigma(n):=\sum_{d|n}d$ be the sum of all divisors of $n$. Find the asymptotic formula for $\sum_{n\leq x}\frac{\sigma(n)}{n}$ and use it to find the one for $\sum_{n\leq x}\sigma(n)$. Here is ...
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Is it always true that if $\sigma(n) + 1$ is divisible by $n$ then $\sigma(n) = 2n - 1$?

Suppose $n$ is a natural number, such that $\sigma(n) + 1$ is divisible by $n$. Is it always true that $\sigma(n) = 2n - 1$? I checked this for all numbers less than $1000000$ and did not find any ...
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1answer
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What is the common (and simplified) value for $D(q^k)D(n^2) = 2s(q^k)s(n^2)$ when $q^k n^2$ is an odd perfect number?

In an answer to an earlier question, it is shown that $$D(2^p - 1)D(2^{p-1}) = 2s(2^p - 1)s(2^{p-1}) = 2^p - 2,$$ if $2^{p-1}(2^p - 1)$ is an even perfect number, $D(x) = 2x - \sigma(x)$ is the ...
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Is $(q^k n^2 \text{ is perfect }) \iff (D(q^k)D(n^2) = 2s(q^k)s(n^2))$ only true for odd perfect numbers $q^k n^2$?

(Preamble: This question is an offshoot of this earlier MSE post.) The title says it all. Is $\bigg(q^k n^2 \text{ is perfect }\bigg) \iff \bigg(D(q^k)D(n^2) = 2s(q^k)s(n^2)\bigg)$ only true for ...
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1answer
35 views

On a theorem of Zumkeller related to twin primes

Let $\{p_1,p_2\}$ be a twin prime pair, $\phi(n)$ denote Euler's totient function and $\sigma(n)$ the sum-of-divisors function. Reinhard Zumkeller proved in 2002 that $$ \phi(p_2) = \sigma(p_1). $$ ...
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Are there any $\ m\ $- superperfect numbers for $\ \ m>2\ \ $?

Perfect numbers are numbers that have the property $$\sigma(n) = 2n$$ A generalization of perfect numbers are superperfect numbers, which have the property $$\sigma(\sigma(n)) = 2n$$ I wonder if there ...
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Are there any other squares $n^2$ for which $\gcd(n^2, \sigma(n^2)) = 2n^2 - \sigma(n^2)$?

Let $\sigma(x)$ denote the sum of the divisors of the positive integer $x$. Denote the deficiency of $x$ by $$D(x)=2x-\sigma(x).$$ I am interested in solutions to the equation $$\gcd(n^2, \sigma(n^2)...
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Inquiry about the Wolfram MathWorld entry on Odd Perfect Number

I just have a quick inquiry about the Wolfram MathWorld entry on Odd Perfect Number. Last sentence of the fifth paragraph states that: Hagis (1980) showed that odd perfect numbers must have at ...
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1answer
223 views

Can the difference between consecutive even abundant numbers exceed 6?

I came across an astonishing observation : An abundant number is a positive integer $n$ with the property $S(n)>n$ , where $S(n)$ is the sum of the divisors of $n$ except $n$ itself. The ...
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1answer
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What is known about iterative behavior of this function?

Let $p$ be a function that sums divisors of some natural number but which does not sum that number. For example we have $p(12)=1+2+3+4+6=16$. We can take that $p(1)=1$ to avoid possible difficulties ...
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1answer
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Does there exist $a_0$, such that $\{a_n\}_{n=0}^{\infty}$ is unbounded?

Suppose $\{a_n\}_{n=0}^{\infty}$ is a sequence, defined by the recurrence relation $$ a_{n+1} = \phi(a_n) + \sigma(a_n) - a_n, $$ where $\sigma$ denotes the divisor sum function and $\phi$ is Euler'...
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1answer
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Is $\phi(n) + \sigma(n) \geq 2n$ always true? [duplicate]

Suppose $\phi$ is Euler totient function and $\sigma$ is divisor sum. Is $\phi(n) + \sigma(n) \geq 2n$ true for every natural $n$? I manually checked the inequality for all numbers between $1$ and $...
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Question about a result on odd perfect numbers

In the paper titled Improving the Chen and Chen result for odd perfect numbers (Lemma 8, page 7), Broughan et al. show that if $$\frac{\sigma(n^2)}{q^k}$$ is a square, where $\sigma(x)$ is the sum of ...
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1answer
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Are all numbers of the type $\frac{n!}{2}+1$ deficient?

Are all numbers of the type $\frac{n!}{2}+1$ deficient? Deficient numbers are such numbers $k$, that the divisor sum of $k$ is less than $2k$. I checked all numbers of this type, with $n$ ranging ...
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What are the properties of abundancy numbers?

Define abundancy numbers as the rational numbers that are equal to the abundancy index of some integer (not to be confused with «abundant numbers», which are natural numbers with abundancyindex ...
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Can a composite number $n$ be the arithmetic mean of $\sigma(n)$ and $\varphi(n)$?

Let $\varphi(n)$ be the totient-function $\sigma(n)$ be the divisor-sum-function It is clear that every prime number $n$ is the arithmetic mean of $\varphi(n)$ and $\sigma(n)$, in other words , ...
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Solutions of the equation $(m! + 2)\sigma(n) = 2n \cdot m!$ where $5 \leq m$

Are there any pairs of natural numbers $(m, n)$, with $5 \leq m$, other than $(5, 15128)$ and $(6, 366776)$, that satisfy the condition $(m! + 2)\sigma(n) = 2n \cdot m!$, where $\sigma(n)$ denotes the ...
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1answer
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On the quantity ${n^2}/D(n^2)$ where $n^2$ is the non-Euler part of members of the OEIS sequence A228059

Let $\sigma(x)$ denote the sum of the divisors of the number $x \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $x$ as $D(x):=2x-\sigma(x)$. This afternoon I noticed some ...
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1answer
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Is $198585576189$ a member of OEIS sequence A228059?

I currently do not have enough computing power, so please pardon me for my question, which occurred just recently to me. So here it goes: Is the Descartes spoof $$\mathscr{D} = {3^2}\cdot{7^2}\...
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Specific evidence for or against the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd ...
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1answer
53 views

Do positive integers $a,b,c,d$ exist with the given properties?

Inspired by this question : Amicable pairs of numbers and their product I ask whether positive integers $a,b,c,d$ exist with the following properties : $(1)\ \ \ \ \ 0<a<b<c<d\ \ \ $ $(...
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3answers
585 views

Are there any natural numbers $n$ that satisfy the condition $7921\sigma(n) = 15840n$?

Are there any natural numbers $n$ that satisfy the condition $7921\sigma(n) = 15840n$, where $\sigma(n)$ denotes the sum of divisors of $n$? This question arises from the theory of immaculate groups (...
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1answer
60 views

Is ${n^2}/D(n^2) \in \mathbb{N}$, if $q^k n^2$ is an odd perfect number?

Let $x \in \mathbb{N}$, the set of positive integers. The sum of the divisors of $x$ is denoted by $\sigma(x)$. Denote the deficiency of $x$ by $D(x):=2x-\sigma(x)$, and the sum of the aliquot parts ...
0
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1answer
50 views

Is 31 the only number that can be represented by two distinct sums of consecutive powers of primes? [duplicate]

I'm trying to prove that a number with two distinct prime factors can't be friends with another number with the same prime factors. One way I could prove this is that there'd be only one example ...
0
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1answer
39 views

Prove this factorial fractional equivalence

I'm supposed to prove the following: $$ \sum_{i=0}^{n-1}\dfrac{(2n-2-i)!}{(n-1-i)!} = \sum_{i=0}^{n-1} \dfrac{(n+i-1)!}{i!}$$ I would greatly appreciate it if people could please take the time to ...
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0answers
42 views

Around the equation $4\,\varphi \left(2^{n-1} \sigma(n)\right) ^2=2^{\sigma(n)}\cdot\varphi(n)^2$

Here for integers $m\geq 1$ I denote the Euler's totient function as $\varphi(m)$ and the sum of positive divisors $\sum_{d\mid m}d$, as $\sigma(m)$. I know that it is possible to prove the following ...
4
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1answer
282 views

Is it possible to simplify this expression even further?

(Preamble: This question is tangentially related to this earlier one.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $...
3
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1answer
61 views

$\phi(n)^{\sigma(n)^{\tau(n)}}=n^2$ find all natural numbers $n$ such that the equality is true

I found this problem in a old number theory test about arithmetic functions. The problem says that a number $n \in N$ is "perfectly crazy" if $$\phi(n)^{\sigma(n)^{\tau(n)}}=n^2,$$ and, as an example, ...