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Updated Oct 8

Problem 410

Statement. 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 all $n\geq 2$

\lim_{k\to \infty} \sigma_k(n)^{1/k}=\infty?

Status. Open.

Source. erdosproblems.com/410, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #410, https://www.erdosproblems.com/410.

References.

[Gu04] Guy, Richard K., Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp.; B41 "Iterations of $\phi$ and $\sigma$ ", printed p. 148: the third of six statements on the iterates $\sigma^k(n)$ that [EGPS90] "are unable to prove or disprove", "for every $n>1$ , $(\sigma^k(n))^{1/k}\to\infty$ as $k\to\infty$ ?". Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (14)

Arithmetic Functions Arithmetic Functions arithmetic_functions/cohen_1996_iterating_sum_divisors_function Remark (p. 98, unnumbered): evidence for statement (iii) and its link to statement (ii)arithmetic_functions/erdos_1990_normal_behavior_iterates_arithmetic_functions Statements (i)–(vi) (p. 169): six assertions on iterates of σ the authors can neither prove nor disprove arithmetic_functions/maier_1984_third_iterates_phi_sigma_functions arithmetic_functions/pollack_2016_problems_erdos_sum_divisors_function Theorem 1.2 (p. 2): up-down aliquot reversals in [1,x] number at least a constant times x/((log_2 x)(log_3 x)^3)Theorem 1.3 (p. 2): down-up aliquot reversals in [1,x] number at least a constant times x/((log_2 x)(log_3 x)^2)arithmetic_functions/pomerance_2018_first_function_iterates Theorem 1.1 (p. 2): the second aliquot step has the Bosma–Kane average Theorem 2.4 (p. 5): conditional averages of log(s_k/s_{k-1}) on density-one sets number_theory/guy_2004_unsolved_problems_number_theory

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