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Claim. For $a=2$ and $p=2$ the exponents $k$ with $2^k\mid 2!+a_2!+\cdots+a_n!$ for some $2<a_2<\cdots<a_n$ are bounded, the first question of Problem 404 at this pair, and $f(2,2)\le254$. The site's commentary credits the bound to S. Lin, On two problems of Erdős concerning sums of distinct factorials, a Bell Laboratories internal memorandum cited as [Li76], the same memorandum whose finiteness theorem for powers of two is recorded on Problem 403's claim page. The site's commentary on Problem 403 states the memorandum's result in the form that $2^{254}$ is the largest power of $2$ that can divide a sum of distinct factorials one of which is $2!$, which is the equality $f(2,2)=254$; the matching lower bound, an explicit sum of $119$ distinct factorials divisible by $2^{254}$, is recorded on Kitamura's page.

Covers. The first question at $a=2$, $p=2$: a finite bound exists, with $f(2,2)\le254$. Not covered: any other pair $(a,p)$, the behavior of $f$, and the third question.

Depends on. No page of this wiki.

Posting and date. The memorandum has no known public copy; the site's problem page (last edited 29 September 2025) is the only link. The monograph of Erdős and Graham cites it as [Lin (76)], a memorandum of 1976, and the site's key agrees, while the site's citation prints 1960, a misprint; the page takes the year 1976 and, knowing no month, is dated to the first of January 1976.

Standing. Claimed: an unrefereed internal memorandum with no public copy, credited in the site's commentary on a problem the site labels OPEN, which is not acceptance. The claim stays claimed.