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 all $n \geq 2$ $k \to \infty lim σ_{k} (n)^{1/ k} = \infty ?$

Worked, still open.

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

machinery: iterated-arithmetic-function,sum-of-divisors-sigma,abundancy-ratio-bound,normal-order-small-prime-factors,anatomy-of-integers,EGPS-iteration-framework,prime-distribution

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

* If you allow **(n=1)**, then it is **false**: (\sigma(1)=1), so (\sigma_k(1)=1) for all $k$ and (\sigma_k(1)^{1/k}=1).

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_410 : answer(sorry) ↔ ∀ n > 1,
    Tendsto (fun k : ℕ ↦ ((sigma 1)^[k] n : ℝ) ^ (1 / (k : ℝ))) atTop atTop

formal-conjectures/410.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

links

#412Let

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)

?A007497

status