# Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

507 questions
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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 $\sigma(n)/n = 5/3$, 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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### 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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### 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 $m$ 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. ...