# Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

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### Does the following GCD divisibility constraint imply that $\sigma(m^2)/p^k \mid m$, if $p^k m^2$ is an odd perfect number with special prime $p$?

The topic of odd perfect numbers likely needs no introduction. In what follows, denote the classical sum of divisors of the positive integer $x$ by $$\sigma(x)=\sigma_1(x).$$ Let $p^k m^2$ be an odd ...
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### Improving $I(m^2)/I(m) < 2^{\log(13/12)/\log(13/9)}$ where $p^k m^2$ is an odd perfect number with special prime $p$

In what follows, let $I(x)=\sigma(x)/x$ denote the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$. The following is an attempt to ...
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### Upper bounds on the greatest common divisor of sums of geometric series

Let $S_1=\sum_{i=0}^{n} p^i = \frac{p^{n+1}-1}{p-1}$ and $S_2=\sum_{i=0}^{m} q^i = \frac{q^{n+1}-1}{q-1}$ be two sums of geometric series, and $\gcd\left(S_1,S_2\right)$ its greatest common divisor. ...
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### Solutions of equations of form $\tau(n+a_i)=\tau(x+a_i)$ where $\tau$ is the number of divisors function and $a_i$ is a diverging series

Consider the function $$\tau(n)=\sum_{d|n}1$$ which gives the number of divisors of a number. The question is: How much information does $\tau(n)$ contain about $n$? The answer is obviously: not very ...
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### The number of positive divisors of a number that are not present in another number.

How many positive divisors are there of $30^{2024}$ which are not divisors of $20^{2021}$? I have tried many ways to try to get a pattern for this problem but I can't. I know that $30$ has $8$ ...
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### Maxima of the Sum of Divisors Function and Upper Bound of a similar Ratio

If we define a function (aka Gronwall’s function) as: $$F(n)=\frac{\sigma(n)}{n \log \log n}$$ Then for $n>15$, it does have an upper bound. I want to know what's that specific upper bound is? Also ...
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1 vote
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### Can the sum of odd divisors of an integer exceed n [closed]

I know that the sum of divisors of an integer n can exceed it (abundant numbers) but can this occur when only considering odd divisors of n? Can the sum of all integers j : j|n, 2∤j be greater than n? ...
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### Prime numbers which end with $59$ or $79$ [closed]

This is the weak conjecture of https://math.stackexchange.com/questions/4834936/prime-numbers-which-end-with-19-39-59-79-or-99.\ $\varphi(n)$ denotes the Euler’s totient function, $n$ denotes a ...
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### A question about prime numbers, totient function $\phi(n)$ and sum of divisors function $\sigma(n)$

I noticed something interesting with the totient function $\phi(n)$ and sum of divisors function $\sigma(n)$ when $n > 1$. It seems than : $\sigma(4n^2-1) \equiv 0 \pmod{\phi(2n^2)}$ only if ...
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1 vote
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### Prove that there are infinitely many natural number such that $σ(n)>100n$

The problem is as follows: Prove that there are infinitely many natural numbers such that $σ(n)>100n$. $σ(n)$ is the sum of all natural divisors of $n$ (e.g. $σ(6)=1+2+3+6=12$). I have come to the ...
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### Continued aliquot sums

What happens if one takes the aliquot sum of an integer and then repeats the process so that one takes the aliquot sum of all of those factors that were not reduced to the number 1 on the previous ...