Erdős Problem #400

The problem

For any $k\geq 2$ let $g_k(n)$ denote the maximum value of

$$(a_1+\cdots+a_k)-n$$

where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid n!$. Can one show that

$$\sum_{n\leq x}g_k(n) \sim c_k x\log x$$

for some constant $c_k$? Is it true that there is a constant $c_k$ such that for almost all $n<x$ we have

$$g_k(n)=c_k\log x+o(\log x)?$$

Statement quoted from erdosproblems.com/400, maintained by Thomas Bloom. Original source: [ErGr80,p.77]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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