{"schema":"vela.problem-packet.v0.1","problem":404,"statement":"For which integers $a\\geq 1$ and primes $p$ is there a finite upper bound on those $k$ such that there are $a=a_1<\\cdots<a_n$ with\\[p^k \\mid (a_1!+\\cdots+a_n!)?\\]If $f(a,p)$ is the greatest such $k$, how does this function behave?Is there a prime $p$ and an infinite sequence $a_1<a_2<\\cdots$ such that if $p^{m_k}$ is the highest power of $p$ dividing $\\sum_{i\\leq k}a_i!$ then $m_k\\to \\infty$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}