Erdős Problem #390: the second-order constant for $f(n)-2n$

Let $f(n)$ be the least $m$ for which $n!$ can be written as $a_1\cdots a_k$ with $n < a_1 < \cdots < a_k = m$ - the smallest possible largest factor in a factorization of $n!$ into distinct integers all exceeding $n$. Erdős, Guy and Selfridge proved $f(n) - 2n \asymp n/\log n$. Erdős asked whether there is a constant $c$ with $$f(n) - 2n \sim c\,\frac{n}{\log n},$$ and what it is. This preprint answers yes and names the constant: $$\lim_{n\to\infty}\frac{(f(n)-2n)\log n}{n} = \frac{4029639598}{25970038185} \approx 0.15516.$$

The headline is the constant, and its two halves have different histories. The lower bound, $\liminf (f(n)-2n)/(n/\log n) \ge 4029639598/25970038185$, is not new here: it is Mausberg's thirteen-layer valuation cut, posted to the erdosproblems.com forum in May 2026 and credited as such in the paper. Its author wrote there that it "does not prove an upper bound, nor does it prove that an asymptotic constant exists." The novelty is the matching upper bound, so the claim is that the thirteen-layer bound is exactly tight. It is assembled from an exact cofactor-allocation certificate, central-binomial anchors, a guarded rough-signature selector, a friable-number covariance bridge, a finite-band tangent correction, and column-sparse rounding. That construction is what a reader should scrutinize; everything else is inherited or machine-checked. erdosproblems.com still lists #390 as open, and the paper calls itself a proposed solution.