How many solutions are there to $σ (a) + σ (b) = σ (a + b)$ with $a + b \leq x$ , where $σ$ is the sum of divisors function? Is it $\sim c x$ for some constant $c > 0$ ?

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

* **Concrete / proved:** There is a **positive linear lower bound**. One can rigorously show [ N(x)\ge c,x ] for some explicit (c>0) [[nomath]](for example $c=2/9$ from the family $(n,2n)$ with $(n,6)=1$)[[/nomath]].

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_1061 : answer(sorry) ↔ ∃ c : ℝ, 0 < c ∧ S ~[atTop] (fun x : ℝ ↦ c * x)

formal-conjectures/1061.lean ↗

oeis

A110177 — Number of solutions 0<k<n to the equation sigma(n) = sigma(k) + sigma(n-k), where sigma is the sum of divisors function.0,0,2,0,0,0,0,2,2,2,0,0,0,0,2,0,0,0,0,2,2,0,0,0,0,0,0,0,0,2,0,4,2,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,2,0,2,0,0,0,4,2,4,0,0,0,

status

open

evidence

formal

oeis

status

Search Vela