Let ${\sigma }_1(n)=\sigma (n)$, the sum of divisors function, and ${\sigma }_k(n)=\sigma ({\sigma }_{k-1}(n))$. Is it true that, for every $m,n\ge 2$, there exist some $i,j$ such that ${\sigma }_i(m)={\sigma }_j(n)$?

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

* For every (n>1) we have (\sigma(n)\ge 1+n), hence (\sigma(n)>n). So each forward orbit [ n,\ \sigma(n),\ \sigma_2(n),\ \sigma_3(n),\dots ] is a **strictly increasing** infinite sequence. * Therefore, if (\sigma_i(m)=\sigma_j(n)) ever happens, then the two orbits **merge forever after** (all subsequent iterates are eq…

llm-hunter · gpt pro 5.2 · unverified

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