Erdős problem 392 · problems.science

Let $A(n)$ denote the least value of $t$ such that $n! = a_1 \cdots a_t$ with $a_1 \leq \cdots \leq a_t\leq n^2$ . Then $A(n) = \frac{n}{2} - \frac{n}{2\log n} + o\left(\frac{n}{\log n}\right).$

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FormalConjectures/ErdosProblems/

392.lean

Retained formal statement2 of 3

Cambie has observed that a positive answer follows from the result above with $a_t \leq n$ , simply by pairing variables together, e.g. taking $a'_i = a_{2i-1}a_{2i}$ (and the lower bound follows from Stirling's approximation).

FormalConjectures/ErdosProblems/392.leanErdos392.erdos_392.variants.implication8 linesExact file

1(∀ (A : ℕ → ℕ),2    (∀ n > 0,3        IsLeast {x | ∃ t, ∃ (_ : ∃ a, n.factorial = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n ^ 2), t + 1 = x} (A n)) →4      (fun n => ↑(A n) - ↑n / 2 + ↑n / (2 * Real.log ↑n)) =o[Filter.atTop] fun n => ↑n / Real.log ↑n) →5  ∀ (A : ℕ → ℕ),6    (∀ n > 0,7        IsLeast {x | ∃ t, ∃ (_ : ∃ a, n.factorial = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n), t + 1 = x} (A n)) →8      (fun n => ↑(A n) - ↑n + ↑n / Real.log ↑n) =o[Filter.atTop] fun n => ↑n / Real.log ↑n

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