List questions from experimental-mathematics | MathQa

Limit superior, limit inferior and a series involging $\sum_{k\nmid n}$k, where $1\leq k\leq n$

Published on 14 Apr 2025 - 8:07

On $\sigma(n)<n+\frac{n}{2}\log\frac{n+1}{n-1}+\frac{n}{4} \left(\sum_{\substack{d|n,1<d<n}}\log\frac{(n+d)(n+1)}{(n-d)(n-1)}\right)$

Published on 14 Apr 2025 - 9:18

With $rad(N)=\prod_{p|N}p$, if $N$ is even and $\frac{2+rad(N)}{8}\left(\sum_{\substack{d|N,d<rad(N)}}d\right)=N$ then is perfect?

Published on 14 Apr 2025 - 11:05

If $n$ satisfies $\left(-3+\sqrt{1+8n}\right)\sigma(n)=4\left(-1+\sqrt{1+8n}\right)\phi(n)$ then is an even perfect number?

Published on 14 Apr 2025 - 15:05

With $s(n)=\sum_{k=1}^n n \bmod k$, can be justified that $\forall\epsilon>0$ let us $\lim_{n\to\infty}\frac{s(n-1)}{\epsilon+s(n)}=1?$

Published on 14 Apr 2025 - 15:06

Hints to compute if exists $\lim_{n\to\infty}\sum_{k=1}^n\sigma(k^2)/\sum_{k=1}^n\sigma(k)$, which $\sigma(n)=\sum_{d\mid n}d$, and other question

Published on 14 Apr 2025 - 8:12

Please, help to identify this numerical constant

Published on 14 Apr 2025 - 7:19

Trilogarithm $\operatorname{Li}_3(z)$ and the imaginary golden ratio $i\,\phi$

Published on 13 Apr 2025 - 13:19

For $n\geq 1$, $\sum_{k=0}^{\infty}\frac{(-1)^{k}(nk+1)^{3}}{(k+1)^6}$ in terms of $\zeta(3)$ and $\zeta(5)$ from a series calculator. Is possible?

Published on 14 Apr 2025 - 8:23

Conjecture $\int_0^1\ln\ln\left(\frac{1+x}{1-x}\right)\frac{\ln x}{1-x^2}\,dx\stackrel?=\frac{\pi^2}{24}\,\ln\left(\frac{A^{36}}{16\,\pi^3}\right)$

Published on 14 Apr 2025 - 8:12