Let $\text{bump}(n,k)$ be a function that increases the prime index of each prime factor of $n$ (with multiplicity) by $k$. e.g. $\text{bump}(20,2)=\text{bump}(2^2\cdot 5,2)=\text{bump}(p_1^2\cdot p_3,2)=p_{1+2}^2\cdot p_{3+2}=5^2\cdot 11=275.$ Conjecture Let $S=\{s \in \mathbb N : 2 \leq s \leq ...

Intuitively, the 2-adic isometry $T$ is just a binary string into which we write a $1$ if a Collatz sequence hits an odd number and write a $0$ if hits an even number. But formally: Let $c(x)=(3x+1)/2$ if $x$ is odd and $x/2$ if $x$ is even. Let $c^n(x)$ indicate $n$ compositions of $c$. Then le...