Let $sd(x)$ be the sum of the digits of $x$, and define $a(x)$ as the sequence with $a(x)_0=x$ and $a(x)_{n+1}=a(x)_n+sd(a(x)_n)$.
Say two sequences $(b_n)_{n\in\mathbb N}$ and $(c_n)_{n\in\mathbb N}$ are asymptotically equivalent iff there exists $N\in\mathbb N$ s.t. for every $n\in\mathbb N$, if $n>N$ then $b_n=c_n$.
Are any of $a(1)$, $a(3)$, and $a(9)$ asymptotically equivalent?
For every $n\in\mathbb N$ other than $n=0$, is $a(n)$ asymptotically equivalent to at least one of $a(1)$, $a(3)$, and $a(9)$?
Tested them for a bunch of values and they seemed to be true but I've no idea about showing it
Another possibly interesting question is how far you have to go into $a(n)$ before it aligns with one of the other three, as it seems to always be very low.
@SimplyBeautifulArt Whoa! I hope you enjoyed a bit of a break?!
@SimplyBeautifulArt I'll make a point of stopping in to your chatroom more regularly; I'd love to hear more about your direction and plans. But very happy to see you keeping in touch with us, and contributing!
@SimplyBeautifulArt Curious recursion, though asymptotic equivalence is usually defined as $\lim\limits_{n\to\infty}\frac{b_n}{c_n}=1$ (for example Hazarika, 2015)
Maybe it's more that $d_n:=b_n-c_n$ is an integer Cauchy sequence $\to0$
