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 numbers or positive integers. Recall that a Descartes number is an odd number $n = km$, with $1 < k$, $1 < m$, satisfying $$\sigma(k)(m+1)=2km.$$ ($m$ is called the ...

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 to around 1650 BC contains a table of representations of $2/n$ as Egyptian fractions for odd $n$...

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 numbers or positive integers. Recall that a Descartes number is an odd number $n = km$, with $1 < k$, $1 < m$, satisfying $$\sigma(k)(m+1)=2km.$$ ($m$ is called the ...

(Note: This question is tangentially related to this later 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 numbers or positive integers. Recall that a Descartes number is an odd number $n = km$, with $1 < k...

Let $n$ be a squarefree composite number. Can $\ \sigma(n)+\varphi(n)\ $ be disivible by $\ n\ $ ? $\ \varphi(n)\ $ is the totient function and $\ \sigma(n)\ $ the sum of the positive divisors of $\ n\ $. I solved the case with no more than $\ 3\ $ prime factors. Also, I checked all square...

As requested by Mathphile, since there have been efforts but no complete solutions to some questions raised when this question was asked on MSE, and since we think that here the question is more probable to be solved completely, I reproduce it here, in a different style. The function $$r(b)=\sum...

Most probably $\gcd(a_i,b_i)=1$ and $a_i \geq 1$ and $b_i$ to be odd for $i=1,...,k$ is enough. Since $\{\varphi(n): n \in \mathbb N\}$ can be partitioned into countably many subsets $C_l=\{\varphi(w_{rl}): r \in \mathbb N\}$ such that $\varphi(w_{ml})<\varphi(w_{nl})$ if $m<n$ the $k$ forms $a...

I'm trying to compute the contour integral $$\frac{1}{2 \pi i} \int_{c - i \infty}^{c + i \infty} \zeta^2(\omega) \frac{8^\omega}{\omega} \ d \omega$$ where $c > 1$, $\zeta(s)$ is the Riemann zeta function. Using Perron's Formula and defining $D(x) = \sum_{k \leq x} \sigma_0(n)$, where $\sigma_0...

I was doing some fairly simple research a few hours ago and I almost asked a similar question with the word continuous instead of differentiable in the title, but then I found this question asked by Gro-Tsen where there is an affirmative answer to that question. Apparently, that is the result of...

At least theory of numbers is full of open problems, if you really much want to solve some of them, or at least one. But if you lost guidance of some professor that does not mean that you lost your own "guidance of yourself", so, when you see that you can "smoothly" read some topic that really i...

Suppose that the quadratic $x^2+bx+c$ has integer coefficients and non-zero discriminant and constant term. Then it is a simple exercise for students to prove that the number of positive integer roots of $x^2+bx+c$ added to the number of positive integer roots of $x^2-bx+c$ is either 0 or 2. Thi...

Suppose that the quadratic $x^2+bx+c$ has integer coefficients and non-zero discriminant and constant term. Then it is a simple exercise for students to prove that the number of positive integer roots of $x^2+bx+c$ added to the number of positive integer roots of $x^2-bx+c$ is either 0 or 2. Thi...

Although Szeto's answer: $$\int x^{x^{...}} dx= \sum^{\infty}_{n=0}\frac{(-n)^{n-1}}{n!}\gamma(n,-\ln(x))$$ is correct. It does not converge for the complete domain of $x^{x^{x...}}$ which is $e^{-e}\le x \le e^{1/e}$ and instead only converges for $e^{-1/e}\le x\le e^{1/e}$. This is because the ...