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I'd like to sum up some specific parts of Wojowu's answer: The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$ $\varphi(n)/n < 1/2$ also for $3 \cdot …

[Please allow me a note: The way desribed below allows to depict functions $f:X^2 \rightarrow Y$ completely in two dimensions (without hiding or omitting any information). This allows for depicting fu …

Initially, I wanted to ask this question as a puzzle. Consider a regular $m$-gon. Let $0$ be the lower corner and count the corners clockwise. Let $n_m$ be the multiplication-by-$n$-graph of …

When coloring the squares of the Ulam spiral not only by black and white (for being prime or non-prime) but by shades of grey representing the normalized totient function $\varphi(n)/n$ and display …

Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$. These graphs often look s …

I investigated the evolution of a single black cell on 1-dimensional grids with periodic boundary conditions of variable sizes $N$ under Wolfram's rule 110 which is the only one for which Turing compl …

I investigated the long term behaviour of a pair of black cells ■■ on a circle of $N$ cells under the action of each of Wolfram's rules $R$. For each combination $(R,N)$ I determined the first occurre …