# Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

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### The sum of divisor function

In the book of Tom M. Apostol introduction to analytic number theory in the prove of theorem 3.3 the sum of divisor function $d(n)$ $$\sum_{n<x}^{}1 =\sum_{n<x}^{}\sum_{d|n} 1$$ He said that ...
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### Is there a number $\mathscr{D}_2 \neq \mathscr{D} = {{3003}^2}\cdot{22021}$ satisfying a certain condition?

(Note: This question is tangentially related to this earlier one.) Let $$\sigma(x) = \sum_{d \mid x}{d}$$ denote the sum of divisors of $x \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural ...
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### On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\operatorname{rad}(n)$, on assumption that $n$ is an odd perfect number

I don't know if my question can be answered easily from your reasonings and knowledeges of the theory of odd perfect numbers. I wondered about it yesterday. Now this post is cross-posted on ...
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### Can we estimate the divisor function?

I have a question about the estimate of the divisor function. Let $$d(n)=\sum_{d|n}1.$$ I proved that $$\sum_{n<x}d(n) \ll x \log x \\ \sum_{n<x}\frac{d(n)}{n} \ll (\log x)^2.$$ My question ...
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### Why did the Egyptians not represent $2/3$ as a sum of unit fractions in the Rhind papyrus?

The following is taken verbatim from the MathWorld Wolfram page on Egyptian fractions: An Egyptian fraction is a sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated ...
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### Divisibility of odd numbers and its sum of divisors function - Part II

This question is inspired by this earlier one: Divisibility of odd numbers and its sum of divisors function In that question, MSE user Juan Moreno claims to have discovered a proof for the following ...
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### On bounds for the deficiency of $m^2$, where $p^k m^2$ is an odd perfect number with special prime $p$

Hereinafter, call a number $N$ perfect if $N$ satisfies $\sigma(N)=2N$, where $$\sigma(x)=\sum_{d \mid x}{d}$$ is the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by ...
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### An improved inequality for the deficiency function when $\gcd(x,y)=1$, $x > 1$, and $y > 1$

(The following is an attempt to improve on the result contained in this MSE question.) Let $\sigma(x)$ be the sum of the divisors of a (positive) integer $x$. (For example, $\sigma(2) = 1 + 2 = 3$.) ...
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### An inequality for the sum-of-aliquot-divisors function

In what follows, we shall assume that $a$ and $b$ are relatively prime. (That is, $\gcd(a,b)=1$ holds.) It is known that the inequality $$\sigma(ab) \leq \sigma(a)\sigma(b)$$ holds for the sum-of-...
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### The equation $\sigma(n)=\sigma(n+1)$

In OEIS, the solutions of $$\sigma(n)=\sigma(n+1)$$ where $\sigma(n)$ denotes the sum of the divisors of $n$ including $1$ and $n$ , are shown upto $n=10^{13}$ The entry can be found already by ...
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### On certain inequalities involving the sum of divisors function and the main branch of the Lambert $W$ function

In this post we denote the main/principal branch of the Lambert $W$ function as $W(x)$ and the sum of divisors function $\sum_{1\leq d\mid n}d$ as $\sigma(n)$. I add as references the articles from ...
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### Can we decide whether $\sigma(n)-2n=k$ is solvable?

An integer $\ k\$ is given. Can we decide whether $\ \sigma(n)-2n=k\$ has a solution and if yes, can we find one solution efficiently ? The following even numbers were not "covered" by the range ...
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### Ratio of the average of the divisor function $\sigma_k(n)$ for even / odd n

Question: Is there anything known (for example a proof) about the ratio $\frac{\bar{\sigma_k}(even)}{\bar{\sigma_k}(odd)}$ where $\bar{\sigma_k}$ is the average value which the divisor function ...
Background Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. A number $l$ is called perfect if $\sigma(l)=2l$. Let $n$ be an odd perfect number given in the so-called Eulerian ...