# Tagged Questions

For questions on arithmetic functions, a real or complex valued function $f(n)$ defined on the set of natural numbers.

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### Computational experiments around the divisor functions, and the matrix $A= \begin{pmatrix} n & \sigma(n) \\ 1 & \sigma_0(n) \end{pmatrix}$

Let for integers $n\geq 1$ the sequence of matrices defined by $$A= \begin{pmatrix} n & \sigma(n) \\ 1 & \sigma_0(n) \end{pmatrix},$$ where $\sigma(n)=\sum_{d\mid n}d$ is the sum of divisors ...
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### Question about multiplicative arithmetic functions

Let $f,g:\Bbb N\to \Bbb C$ be multiplicative arithmetic functions, i.e. $$\gcd(m,n)=1\implies f(mn)=f(m)f(n)$$ and same for $g$. We can also assume $f(1)\neq 0$ and $g(1)\neq 0$ if necessary. How can ...
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### On a criterion for almost perfect numbers using the abundancy index

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. A number $y$ is said to be almost perfect if $\sigma(y) = 2y - 1$. In a preprint ...
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### Prove that $\bigg\lfloor\frac{m}{p^k}\bigg\rfloor-\bigg\lfloor\frac{n}{p^k}\bigg\rfloor-\bigg\lfloor\frac{m-n}{p^k}\bigg\rfloor$ equal to zero or one

Prove that $\bigg\lfloor\frac{m}{p^k}\bigg\rfloor-\bigg\lfloor\frac{n}{p^k}\bigg\rfloor-\bigg\lfloor\frac{m-n}{p^k}\bigg\rfloor$ equal to zero or one for all $k,m,n\in \mathbb N$. where $m\geqslant n$ ...
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### Perfect numbers of the form $12m+1$ and $\sum_{d\mid n}\frac{1}{\phi(d)}$, where $\phi(m)$ is Euler's totient function

If there are no mistakes combining Exercise 9 a) (Chapter 3, page 71) and Exercise (Chapter2, page 47) of Apostol's Introduction to Analytic Number Theory we can prove easily Lemma. If $n$ is a ...
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### On the density of solitary numbers

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. If $X$ is the unique solution of $$I(X) = \dfrac{a}{b}$$ (for a given rational ...
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### Can you analyze this identity involving the sum of divisors function and $rad(n)=\prod_{p\mid n}p$?

Let $\sigma(n)$ the sum of divisors function, then by Mobius inversion formula $$\sigma(n)=n-\sum_{\substack{d\mid n,d<n}}\sigma(d)\mu\left(\frac{n}{d}\right),$$ and since this function is ...
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### On the indivisibility of odd perfect numbers by small numbers

A good day to everyone! This question is an offshoot of the following MSE posts: Odd perfect number divisors Can an odd perfect number be divisible by $101$? My question is as follows: Is ...
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