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Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

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On the golden ratio and odd perfect numbers

Here is my question: Is $I(n^2) - 1 > 1/I(n^2)$ true when $I(n^2)=\sigma(n^2)/n^2$ is the abundancy index of $n^2$ and $q^k n^2$ is an odd perfect number with special prime $q$ satisfying $k>...
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If $\sigma(n)/n = 5/3$, then $5 \nmid n$. Does it also follow that $3 \nmid \sigma(n)$?

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. If $\sigma(N)=2N$ (equivalently, when $I(N)=2$) then $N$ is called a ...
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A consequence of assuming the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

Let $x$ be a positive integer. Denote the sum of divisors of $x$ by $$\sigma(x) = \sum_{d \mid x}{d},$$ and the deficiency of $x$ by $$D(x) = 2x - \sigma(x).$$ A number $N$ is said to be perfect if $...
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Can these bounds in terms of the abundancy index and deficiency functions be improved for deficient-perfect numbers?

Let $$\sigma(x) = \sum_{e \mid x}{e}$$ denote the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$, and the deficiency of $x$ by $D(x)=2x-\sigma(x)$...
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Either $np$ is even, or $n^2 − p^2$ is a multiple of $8$. [duplicate]

Let $n$ and $p$ be two integers. Show that either $np$ is even, or $n^2 − p^2$ is a multiple of $8$. If either one of $n$ or $p$ is even then $np$ is even and we are done. So let both of them is odd. ...
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Improving the bound for $\sigma(q^k)/q^k$ where $q^k n^2$ is an odd perfect number given in Eulerian form

Let $x$ be a positive integer. (That is, let $x \in \mathbb{N}$.) We denote the sum of divisors of $x$ as $$\sigma(x) = \sum_{d \mid x}{d}.$$ We also denote the abundancy index of $x$ as $I(x)=\...
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If $\sigma(n) = 2n - d$ and $d \mid n$, is it true that $d = \gcd(n,\sigma(n))$?

In what follows, assume that $d > 0$. Let $$\sigma(x)=\sum_{e \mid x}{e}$$ denote the classical sum-of-divisors function, and denote the deficiency of $x \in \mathbb{N}$ by $$D(x)=2x-\sigma(x).$$ ...
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How to simplify $2 q^{\frac{k-1}{2}} n^2 - \sigma(q^{\frac{k-1}{2}})\sigma(n^2)$

Let $k$ be a positive integer satisfying $k \equiv 1 \pmod 4$. Let $x \in \mathbb{N}$. Let $q$ be a prime number. If $$\sigma(x) = \sum_{d \mid x}{d}$$ is the classical sum-of-divisors function, ...
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Truncated sum of divisors bound

I am interested in an upper bound for $$ \sum_{\substack{d|N\\ d>A}}\frac{1}{d^3},$$ in particular, I can get the above to be $$\sum_{\substack{d|N\\ d>A}}\frac{1}{d^3}\ll \frac{\text{exp}\...
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Can this inequality regarding odd perfect numbers be improved?

Let $\sigma(x)$ denote the sum of the divisors of $x$. Denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$. If $N$ is odd and $\sigma(N)=2N$, then $N$ is called an odd perfect number. Euler showed ...
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On some inequalities relating the special/Euler prime and non-Euler part of odd perfect numbers

Let $N$ be an odd (positive) integer. If $\sigma(N)=2N$ where $\sigma(N)$ is the sum of the divisors of $N$, then $N$ is called an odd perfect number. Let $I(N)=\sigma(N)/N$ denote the abundancy ...
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On Basak's “Bounds On Factors Of Odd Perfect Numbers”

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 what follows, we denote the ...
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If $q \equiv k \equiv 1 \pmod 4$, is it necessarily true that $\gcd\bigg(\sigma(q^k),\sigma(q^{(k-1)/2})\bigg)=1$?

Let $\sigma$ denote the classical sum-of-divisors function. In what follows, we let $q$ be a prime number. Here is my question: If $q \equiv k \equiv 1 \pmod 4$, is it necessarily true that $\gcd\...
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On Yanqi Xu's 2016 joint undergraduate math research project with Dr. Judy Holdener at Kenyon College

In what follows, we let $\sigma(X)$ denote the sum of the divisors of the positive integer $X$. Denote the abundancy index of $X$ by $I(X)=\sigma(X)/X$, and the deficiency of $X$ by $D(X)=2X-\sigma(X)...
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About the equation $x^2+y^2+z^2+2t^2=n$

The question The final goal (for this stage of my project) is to get an explicit form for $\phi(n)$. This last one is the number of integer solutions to $x^2+y^2+z^2+2t^2=n$. You may find this $\phi(...
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Are there infinitely many sets of relatively prime numbers with equal number and sum of divisors?

Consider the prime factorization of the numbers $14$ and $15$ : $$14 = 2 \cdot 7 \implies \tau(14) = 2 \cdot 2 = 4 \space ;\space \sigma(14) = 3 \cdot 8 = 24$$ $$15=3 \cdot 5 \implies \tau(15) = 2 \...
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Limiting value of $\frac{\sigma(n)}{n}$

I am having some trouble with the following: Let $\sigma(n)$ be the sum of the positive divisors of $n$, e.g. $\sigma(6)=1+2+3+6=12$. What is the 'expected value' of the abundancy index $\frac{\...
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Is it true that $\gcd(s(p^k), D(p^k)) = 1$?

Let $\sigma(x)$ be the sum of divisors of a positive integer $x$. Define $$s(x):=\sigma(x)-x$$ to be the sum of the aliquot divisors of $x$, and define $$D(x):=2x-\sigma(x)$$ to be the deficiency of $...
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On an interesting assertion in the OeisWiki page on multiply-perfect numbers

The following (interesting) assertion appears in the OeisWiki page on multiply-perfect numbers: ...
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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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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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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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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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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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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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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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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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229 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 ...