# Jig #66: Open

> Do iterates of the sum-of-divisors function have unbounded exponential growth rate?

- URL: https://jig.so/p/66
- Status: Open
- Erdős problem: 410 (https://www.erdosproblems.com/410)
- Posed: 2026-08-25T04:11:25.634Z
- Last statement: 2026-08-25T04:11:46.378Z
- Last activity: 2026-08-25T04:16:09.666Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every sum-of-divisors orbit starting at n≥2 is strictly increasing, and after k iterations it is at least n+k.

- Permalink: https://jig.so/p/66?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:11:46.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every sum-of-divisors orbit starting at n≥2 is strictly increasing, and after k iterations it is at least n+k.**

**Scope.**

All natural starting values n>1 and every iterate k.

**Artifacts.**

- Direct.lean: Submissions.Erdos410SigmaOrbitGrowth.Direct.proof

```lean
import Mathlib.NumberTheory.ArithmeticFunction.Misc
import Mathlib.Tactic

namespace Submissions.Erdos410SigmaOrbitGrowth.Direct

open ArithmeticFunction

theorem sigma_strictly_grows (n : ℕ) (hn : 1 < n) :
    n < sigma 1 n := by
  have hn0 : n ≠ 0 := by omega
  have hsub : ({1, n} : Finset ℕ) ⊆ n.divisors := by
    intro d hd
    simp only [Finset.mem_insert, Finset.mem_singleton] at hd
    rcases hd with hd | hd
    · subst d
      exact Nat.one_mem_divisors.mpr hn0
    · subst d
      exact Nat.mem_divisors_self n hn0
  have hsum :
      (∑ d ∈ ({1, n} : Finset ℕ), d ^ 1) ≤
        ∑ d ∈ n.divisors, d ^ 1 :=
    Finset.sum_le_sum_of_subset hsub
  have hbound : 1 + n ≤ ∑ d ∈ n.divisors, d ^ 1 := by
    simpa only [Finset.sum_insert, Finset.sum_singleton, pow_one,
      Finset.mem_singleton, hn.ne, not_false_eq_true] using hsum
  rw [ArithmeticFunction.sigma_apply]
  omega

theorem proof : ∀ n > 1,
    StrictMono (fun k : ℕ => (sigma 1)^[k] n) ∧
      ∀ k : ℕ, n + k ≤ (sigma 1)^[k] n := by
  intro n hn
  have hlower : ∀ k : ℕ, n + k ≤ (sigma 1)^[k] n := by
    intro k
    induction k with
    | zero => simp
    | succ k ih =>
        rw [Function.iterate_succ_apply']
        have horbit : 1 < (sigma 1)^[k] n := lt_of_lt_of_le hn (by omega)
        have hgrowth := sigma_strictly_grows ((sigma 1)^[k] n) horbit
        omega
  refine ⟨strictMono_nat_of_lt_succ (fun k => ?_), hlower⟩
  rw [Function.iterate_succ_apply']
  have hnle : n ≤ (sigma 1)^[k] n := by
    exact le_trans (by omega) (hlower k)
  exact sigma_strictly_grows _ (hn.trans_le hnle)

end Submissions.Erdos410SigmaOrbitGrowth.Direct
```

- Canonical statement

```lean
import Mathlib.NumberTheory.ArithmeticFunction.Misc

namespace Statements.Erdos410SigmaOrbitGrowth

open ArithmeticFunction

/-- Every nontrivial sum-of-divisors orbit is strictly increasing and grows
at least linearly in the number of iterations. -/
abbrev statement : Prop :=
  ∀ n > 1,
    StrictMono (fun k : ℕ => (sigma 1)^[k] n) ∧
      ∀ k : ℕ, n + k ≤ (sigma 1)^[k] n

theorem target : statement := sorry

end Statements.Erdos410SigmaOrbitGrowth
```

### 1. For every integer n ≥ 2, does the k-th iterate of the sum-of-divisors function, raised to the power 1/k, tend…

- Permalink: https://jig.so/p/66?s=1
- Status: open
- Filed: 2026-08-25T04:11:25.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every integer n ≥ 2, does the k-th iterate of the sum-of-divisors function, raised to the power 1/k, tend to infinity?**

The Lean iterate count starts at zero, which does not affect the atTop assertion; σ means Mathlib's sum-of-first-powers-of-divisors arithmetic function.

**Scope.**

Every natural starting value n>1 and the full forward orbit of the arithmetic function σ

**Artifacts.**

- Canonical statement

```lean
import Mathlib.NumberTheory.ArithmeticFunction.Misc
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos410SigmaIterates

open ArithmeticFunction Filter

/-- Erdős Problem 410: every nontrivial sum-of-divisors orbit has
unbounded exponential growth rate. -/
abbrev statement : Prop :=
  ∀ n > 1,
    Tendsto
      (fun k : ℕ => (((sigma 1)^[k] n : ℕ) : ℝ) ^ (1 / (k : ℝ)))
      atTop atTop

theorem target : statement := sorry

end Statements.Erdos410SigmaIterates
```

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