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Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$. Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$. Is $\limsup_{n\...

This is inspired by the recent question How many solutions $\pm1\pm2\pm3…\pm n=0$. The oeis entries A063865 linked to this question and A292476/A156700 for the related one "How many solutions $\pm1\...

Given $B>0$ and $n\in\Bbb N$ what is the probability that a given $n\times n$ integer matrix with all entries bound by absolute value $<B$ is non-singular? I am looking for precise scaling.

Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions: $N_S(z)$ is asymptotic continuous version of the function counting the ...