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Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

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Does this iterated sequence always end in a finite number of steps to a number which is divisible by a perfect number?

I posted this question at MathOverflow, but then I realized that maybe it is more appropriate to ask it here: Let $f$ be a multiplicative arithmetic function which maps $\mathbb{N}$ to itself, such ...
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Two questions about the dihedral group

First question: 1) Is the sum of subgroup indices of dihedral group with $2n$ elements equal to $\sigma_2(n)+2\cdot \sigma(n)$? Second question: 2) Is $\sigma_2(n)+2\cdot \sigma(n) \le L(H(D_n))$? ...
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Confused as to how this step in a number theory proof is performed

How does this step $$D(q)=\sum_{n=1}^\infty d(n)q^n$$ Become this step? \begin{align} D(q) &=\sum_{n=1}^\infty\sum_{m|n}mq^n=\sum_{m=1}^\infty\sum_{m|n}mq^n \\ &=\sum_{m=1}^\infty\...
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A recursive identity for the sum of divisors

Let $(n,k)=\gcd(n,k)$ and $(n,l,k) = \gcd(\gcd(n,l),k)$, $\sigma(n)=$ sum of divisors of $n$. My question is, how the "ugly" identity, which I can prove it is true, can be "simplified" in presentation?...
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If $N$ is deficient-perfect, under what conditions does this inequality hold?

This question is an offshoot of the following answer to a closely related MSE question. Let $N$ be a deficient-perfect number, i.e. $N$ is a positive integer such that $D(N) \mid N$ where $D(N)=2N-\...
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Evaluating the sum $\sum_{i=1}^n i^2\cdot\lfloor{\frac ni}\rfloor$

I need to evaluate the sum $$\sum_{i=1}^n i^2\cdot\lfloor{\frac ni}\rfloor$$ After a little bit of math I found that the above sum is equal to: $$\sum_{i=1}^n i\cdot n - \sum_{i=1}^ni\cdot (n\space ...
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Estimating $\sum\limits_{d\mid n}{d+a\choose b}$

Is there any way of estimating a sum like $$\sum_{d\mid n}{d+a\choose b},$$ for positive integers $a$ and $b$? For example, in the OEIS we find that $$\begin{align*} \sum_{d\mid n}{d+1\choose 2} &...
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Sum of divisors of a square

I was wondering if there is a nice formula for the number of divisors $d$ of a perfect square ($n^2$), such that $\frac{n}{2} <d < n$. For example, for $n = 12$, the divisors $d$ of $12^2=144$ ...
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If $q$ is a Fermat prime, is $\sigma(q^k)/2$ a square if $k \equiv 1 \pmod 4$?

In what follows, let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number of the form $M = 2^{2^m} + 1$ is called a Fermat number. If in addition $M$ is prime, then $M$ is ...
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Generalise of $\sigma_k(P_a^j)+\sigma_k(P^{j+1}_a)?$

Where $P_a$ is a prime number and $\sigma_k(n)$ is divisor function We encountered these two identities: $$\sigma_k(P_a)+\sigma_k(P^2_a)=(P_a^k+1)^2+1\tag1$$ $$\sigma_k(P_a^2)+\sigma_k(P^3_a)=(P_a^...
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Finding the sum of the reciprocals of the positive divisors of a number

Let $d_1, d_2, \dots , d_k$ be all the positive factors of a positive integer $n$ including $1$ and $n$. If $d_1+d_2+ \dots+d_k=72$, then find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\dots+\frac{1}{...
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Sum of all positive divisors are prime

For an integer $n$, let $\sigma (n)$ be the sum of all the positive divisors of $n$. For how many integers $n$ in the inclusive range $[1, 500]$ will $\sigma (n)$ be a prime number? I tried doing ...
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Is this $\zeta(p)=\sum_{k\geq1 ,p\in \mathbb{P}}\frac{1}{{\sigma}_k(p)-1}$ true with sigma is power of sum divisor function?

let ${\sigma}_k(n) =\sum_{d|n} d^k$ is a sum of divisor function , And let ${\sigma}_k(n)$ be the iterating divisor function, We have for every prime $p$ and for every integer $k\geq 1$ : $${\sigma}_k(...
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Breaking the barriers at $q=5$ and $q=13$ for $q^k n^2$ an odd perfect number with special prime $q$

Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. If $\sigma(N)=2N$ and $N$ is odd, then $N$ is called an odd perfect number. The question of existence of odd perfect numbers is the ...
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Improving some known bounds regarding the abundancy index of divisors of odd perfect numbers

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the sum of aliquot ...
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On GCDs and odd perfect numbers

Let $N=q^k n^2$ be an odd perfect number with special prime $q$. The index $i(q)$ of $N$ at the prime $q$ is then equal to $$i(q):=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/2}=\frac{D(n^2)}{s(q^...
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What is $\sigma(q^k)$ in terms of $k$ if $q \equiv 5 \pmod 8$?

Denote the sum of divisors of $x \in \mathbb{N}$ by $\sigma(x)$. Here is my question: What is $\sigma(q^k)$ in terms of $k$ if $q \equiv 5 \pmod 8$? I know that if $q \equiv 1 \pmod 8$, then $$\...
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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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1answer
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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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145 views

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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1answer
67 views

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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30 views

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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59 views

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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1answer
78 views

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 $m$ is 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. ...