OPEN
This is open, and cannot be resolved with a finite computation.
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 $m,n\geq 2$, there exist some $i,j$ such that $\sigma_i(m)=\sigma_j(n)$?
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
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].
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Additional thanks to: Hayato Egami
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #412, https://www.erdosproblems.com/412, accessed 2026-04-11
