The board / number theory / #410
Erdős Problem #410
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(σ_1(n) = σ(n)\), the sum of divisors function, and \(σ_k(n) = σ(σ_{k-1}(n))\). Is it true that \(\lim_{k → ∞} σ_k(n)^{\frac 1 k} = ∞\)? This is problem (iii) from Erdos, Granville, Pomerance, Spiro "On the normal behavior of the iterates of some arithmetical functions" (page 169 of the book "Analytic Number Theory", 1990).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/410.Prizenone
Topicsnumber theory, iterated functions
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA007497
Original pageerdosproblems.com/410
Attempts
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Live page status OPEN (accessed 2026-07-28), with 0 claimed proofs and RomanLeLan listed under “Currently working on this problem”; stopped to avoid colliding with the current worker.
SKIPPED