Problem 412

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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 every $m,n\geq 2$, there exist some $i,j$ such that $\sigma_i(m)=\sigma_j(n)$?

Status. Open.

Source. erdosproblems.com/412, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #412, https://www.erdosproblems.com/412.

References.

• [Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.

Formalization. Statement in formal-conjectures.

Current assessment

ErGr80's attribution, numerical-evidence report and historical outlook on printed p. 81 supply no certified disjoint pair or proof/disproof. The imported Open status is retained, and the linked formal declaration is recorded at statement scope. This page does not date or scope a current literature search or record reviewed proof coverage for the displayed question.

Known Results

Historical formulation and evidence

Erdős and Graham's 1980 monograph, printed p. 81, attributes the iterated-$\sigma$ question to van Wijngaarden in the 1950s. Its literal formulation asks whether, for every $m,n$, some iterates satisfy $\sigma_i(m)=\sigma_j(n)$. It does not state the displayed question's restriction $m,n\geq2$; that question is a strict specialization of the source wording.

The authors report Selfridge's numerical evidence suggesting a negative answer and express doubt that a proof would be available in the near future. This describes the historical evidence and outlook, not a certified counterexample or a current-status argument. Er79d remains the distinct 1979 source already cited above.

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.

• cohen_1996_iterating_sum_divisors_function
• cohen_1996_iterating_sum_divisors_function / conjecture_p98
• cohen_1996_iterating_sum_divisors_function / theorem_3_1
• erdos_1979_unconventional_problems_number_theory
• erdos_1990_normal_behavior_iterates_arithmetic_functions
• erdos_1990_normal_behavior_iterates_arithmetic_functions / statements_p169
• erdos_1980_old_new_problems_results_combinatorial_number_theory