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Claim. For the pairs $(a,p)$ in the repository's exact rows the exponents $k$ with $p^k\mid a_1!+\cdots+a_n!$ for some $a=a_1<\cdots<a_n$ are bounded, the first question of Problem 404 at those pairs, and the largest such exponent is computed: the rows displayed in the thread post are

$$f(3,3)=3,\ f(39,3)=25,\ f(700,3)=363,\ f(110,5)=30,\ f(236,5)=61,$$ $$f(42,7)=7,\ f(113,11)=11,\ f(163,13)=13,$$

and the Lean examples add $f(4,7)=0$ and $f(6,7)=0$. Kenta Kitamura, under the forum name KentaKitamura, announced the repository KitaKen1/erdos-404-odd-prime-landscape in the site's discussion thread on 8 July 2026, the day after the $p=2$ repository recorded on its page; its README (linked at the commit of that day) is the write-up. An exact row means that the search finds a witness at exponent $K$ and rules out exponent $K+1$ by the same finite argument as at $p=2$, with Legendre's formula bounding the candidate terms and a windowed dynamic program over residues modulo $p^{K+1}$. The repository also records witnesses giving $f(1,3)\ge20000$, $f(1,5)\ge20000$, $f(1,7)\ge20000$, $f(1,11)\ge5000$ and $f(1,13)\ge5000$, explicit lists checked modulo $p^K$, in the direction of Tao's thread remark that $f(1,5)$ appears very large and possibly infinite, and baseline lower bounds $f(a,p)\ge v_p(a!)$ for rows the scan left open; the post says these are finite witnesses, not claims of infinitude, and they settle no instance. The three Lean files (the formalization links) prove $f(3,3)=3$, $f(4,7)=0$ and $f(6,7)=0$ from both sides; this corpus has not built them. The post says the search programs, organization, Lean files and the post were prepared with assistance from Codex, ChatGPT and Claude Code (Fable 5); the human submitter is the claimant, with the systems named as the submitter names them.

Submission note. Posted to the site's forum by Kenta Kitamura on 8 July 2026:

I made an attempt repository for Problem #404's function $f(a,p)$, focusing on the odd-prime columns $p=3,5,7,11,13$: Repository: https://github.com/KitaKen1/erdos-404-odd-prime-landscape Visual table: https://kitaken1.github.io/erdos-404-odd-prime-landscape/

Contribution 1: witnesses for the special case $a=1$, i.e. lower bounds for $f(1,p)$. The displayed certificates give $f(1,3)\ge 20000$, $f(1,5)\ge 20000$, $f(1,7)\ge 20000$, $f(1,11)\ge 5000$, and $f(1,13)\ge 5000$. These are finite witnesses, not claims of infinitude; each witness is an explicit increasing list, checked directly modulo $p^K$. On the GitHub Pages visual table, one can click an entry and inspect the concrete list $a_1,\ldots,a_n$.

Contribution 2: rows for the remaining starts $a>1$, i.e. data for $f(a,p)$ beyond the $a=1$ case. Contribution 2-1: exact rows for $a>1$. These mean the search finds a witness at exponent $K$ and rules out exponent $K+1$. Some rows currently displayed are $f(3,3)=3$, $f(39,3)=25$, $f(700,3)=363$, $f(110,5)=30$, $f(236,5)=61$, $f(42,7)=7$, $f(113,11)=11$, and $f(163,13)=13$. Small Lean4Web examples: $f(3,3)=3$: Lean4Web; $f(4,7)=0$: Lean4Web; $f(6,7)=0$: Lean4Web. Contribution 2-2: lower-bound rows for $a>1$. Some cells are marked as scan-survivors. For those cells I am not claiming an exact value. The displayed lower bound is only the baseline certificate from the one-term list $(a)$: it proves $f(a,p)\ge v_p(a!)$. Examples include $f(40,3)\ge18$, $f(300,5)\ge74$, $f(294,7)\ge48$, $f(297,11)\ge29$, and $f(299,13)\ge24$. Whether a larger exponent is possible is left open.

This is in the direction of the discussion around Problem #404, especially Terence Tao's comments about $f(1,5)$ : https://www.erdosproblems.com/forum/thread/404

AI disclosure: the search programs, organization, Lean files, and this comment were prepared with assistance from Codex, ChatGPT, and Claude Code (Fable 5).

Covers. The first question at the pairs $(3,3)$, $(39,3)$, $(700,3)$, $(110,5)$, $(236,5)$, $(42,7)$, $(113,11)$, $(163,13)$, $(4,7)$ and $(6,7)$: a finite bound exists, with the value of $f$ computed exactly. Not covered: the lower-bound rows, which decide nothing; $a=1$ at any odd prime; the behavior of $f$ in general; and the third question.

Depends on. No page of this wiki.

Standing. Claimed: a research note in a public repository, unrefereed, not cited by the site's commentary, with Lean certificates this corpus has not built; the site labels the problem OPEN. The claim stays claimed.