All Random Solved Random Open
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 $m,n$, there exist some $i,j$ such that $\sigma_i(m)=\sigma_j(n)$?
In [Er79d] Erdős attributes this conjecture to van Wijngaarden, who told it to Erdős in the 1950s.

That is, there is (eventually) only one possible sequence that the iterated sum of divisors function can settle on. Selfridge reports numerical evidence which suggests the answer is no, but Erdős and Graham write 'it seems unlikely that anything can be proved about this in the near future'.

See also [413] and [414].