Let $\varphi (n)$ be the Euler totient function and ${\varphi }_k(n)$ be the iterated $\varphi$ function, so that ${\varphi }_1(n)=\varphi (n)$ and ${\varphi }_k(n)=\varphi ({\varphi }_{k-1}(n))$. Let$$f(n)=\min \{k:{\varphi }_k(n)=1\}.$$Does $f(n)/\log n$ have a distribution function? Is $f(n)/\log n$ almost always constant? What can be said about the largest prime factor of ${\varphi }_k(n)$ when, say, $k=\log \log n$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let ( \phi_0(n)=n), (\phi_{k+1}(n)=\phi(\phi_k(n))), and [ f(n)=\min{k:\phi_k(n)=1}. ] This $f(n)$ is the classical “height” (Pillai–Shapiro height) of $n$ under (\phi)-iteration.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.