Let $σ_{1} (n) = σ (n)$ , the sum of divisors function, and $σ_{k} (n) = σ (σ_{k - 1} (n))$ . Is it true that, for every $m, n \geq 2$ , there exist some $i, j$ such that $σ_{i} (m) = σ_{j} (n)$ ?

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

* For every (n>1) we have (\sigma(n)\ge 1+n), hence (\sigma(n)>n). So each forward orbit [ n,\ \sigma(n),\ \sigma_2(n),\ \sigma_3(n),\dots ] is a **strictly increasing** infinite sequence. * Therefore, if (\sigma_i(m)=\sigma_j(n)) ever happens, then the two orbits **merge forever after** (all subsequent iterates are eq…

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_412 : answer(sorry) ↔ ∀ᵉ (m ≥ 2) (n ≥ 2), ∃ i j, (σ 1)^[i] m = (σ 1)^[j] n

formal-conjectures/412.lean ↗

oeis

A007497 — a(1) = 2, a(n) = sigma(a(n-1)).2,3,4,7,8,15,24,60,168,480,1512,4800,15748,28672,65528,122880,393192,1098240,4124736,15605760,50328576,149873152,3712262 A051572 — a(1) = 5, a(n) = sigma(a(n-1)).5,6,12,28,56,120,360,1170,3276,10192,24738,61440,196584,491520,1572840,5433480,20180160,94859856,355532800,1040179456,21

links

#410Let

σ_{1} (n) = σ (n)

, the sum of divisors function, and

σ_{k} (n) = σ (σ_{k - 1} (n))

. Is it true that for all

n \geq 2

k \to \infty lim σ_{k} (n)^{1/ k} = \infty ?

A007497

status