Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
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Is $n^2\mid \sigma(n)^{\sigma(n)}-1$ possible for $n>1$?
Inspired by Martin Hopf I used another number theoretical function for this question.
If $\sigma(n)$ is the divisor-sum function ($1$ and $n$ are included, $\sigma(1)=1$) , can we have $$n^ 2\mid \...
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on the equation $\sigma(n+1)=\sigma(n)+1$
Let $\sigma(k)$ denote the sum of all positive divisors of $k$.
Consider the equation $\sigma(n+1)=\sigma(n)+1$.
Has it been investigated before? (I did some search in books avaliable for me, and in ...
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Estimation of the number of solutions for the equation $\sigma(\varphi(n))=\sigma(\operatorname{rad}(n))$
For integers $n\geq 1$ in this post we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an ...
user243301
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Does there exist an integer $m$ which is not in the image of $D(n)=2n - \sigma(n)$?
Let $\sigma(n)$ be the sum of the divisors of the positive integer $n$, and denote the deficiency of $n$ by $D(n)=2n-\sigma(n)$.
Here is my question:
Do there exist numbers $m$ which (provably) do ...
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Has it been proved that odd perfect numbers cannot be triangular?
(Note: This question has been cross-posted to MO.)
Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...
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Limit superior, limit inferior and a series involging $\sum_{k\nmid n}$k, where $1\leq k\leq n$
The purpose of this post is state assertions by the use of statements and hypothesis in an expository way and after I am asking for reasonable unconditionally results that you can provide us.
Using ...
user243301
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Does there exist a positive $k$ s.t. for all $r\geq k$, "$\sigma_r(m)<\sigma_r(n)$ for every $m<n$" for infinitely many odd positive integers $n$?
Does there exist a positive real $k$ such that for all real $r\geq k$, "$\sigma_r(m)<\sigma_r(n)$ for every $m<n$" for infinitely many odd positive integers $n$? $\sigma_r(n)$ is the sum of the $...
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Can we efficiently check whether a number is a Zumkeller number?
A positive integer $n$ is a Zumkeller number iff its divisors can be partitioned into two sets with equal sum.
If $\sigma(n)$ denotes the divisor-sum-function , this means that there are distinct ...
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Are there infinite many positive integers $\ n\ $ satisfying $\ \varphi(n)|\sigma(n)\ $?
Today I suddenly discovered that many positive integers $\ n\ $ satisfy $\ \varphi(n)\mid \sigma(n)\ $. This leads to the following question :
Are there infinitely many postive integers $\ n\ $ ...
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Fast Computation of the Divisor Summatory Function $\sum_{i=1}^x\sum_{d|i}d$
Denote by $\sigma_1(n)$ the sum of the divisors of $n$. For example, when $n=9$ we get $\sigma_1(9) = 1+3+9=13$. Define $D(x) = \sum_{n=1}^x\sigma_1(n)$. Varius methods are known for computing $D(x)$, ...
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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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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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A mixture with ingredients of two equivalences with Riemann Hypothesis
Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
user243301
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Composite $n$ such that $\sigma(n) \equiv n+1 \pmod{\phi(n)}$
I'm looking for composite $n$ such that
$$\sigma(n)\equiv n+1\pmod{\varphi(n)}$$
Are there only finitely many? Can this be proved?
This is Sloane's A070037 but there's not much information in the ...
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Is $650$ the only solution not fitting in the family?
Inspired by this question
The linked question conjectures that $\frac{\sigma(n)}{n+1}$ (where $\sigma(n)$ denotes the divisor-sum function) is not an integer for any squarefree composite number.
If we ...
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