erdős #825
Is there an absolute constant such that every integer with is the distinct sum of proper divisors of ?
Worked, still open.
number theory · solved · prize $25 · formalized (Lean) · 0 attempts
use this record
vela registry pull vfr_37aec80d874a0239vela reproduce examples/erdos-problemsevidence
unverified AI candidates (2)
gpt-erdos · GPT-5.2 Pro + Deep Research · unverified
What you are asking is exactly the question of whether **weird numbers** can have arbitrarily large “abundancy index.”
candidate solution ↗llm-hunter · gpt pro 5.2 · unverified
1 LLM attack(s) recorded (gpt pro 5.2); unverified.
candidate solution ↗formal
AMS 11 · solved (literature)
theorem erdos_825 :
answer(True) ↔ ∃ (C : ℝ) (_ : C > 0),
∀ (n) (_ : σ 1 n > C * n),
∃ s ⊆ n.properDivisors, n = s.sum idformal-conjectures/825.lean ↗oeis
A006037 — Weird numbers: abundant (A005101) but not pseudoperfect (A005835).70,836,4030,5830,7192,7912,9272,10430,10570,10792,10990,11410,11690,12110,12530,12670,13370,13510,13790,13930,14770,1561A330244 — Weird numbers m (A006037) such that sigma(m)/m > sigma(k)/k for all weird numbers k < m, where sigma(m) is the sum of divisors of m (A000203).70,10430,1554070,5681270,6365870
links
weird numbers · reference
by Larsen · paper
#470Call weird if and is not pseudoperfect, that is, it is not the sum of any set of its divisors.Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of is weird?A006037status
solved