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This is more a curiosity than anything useful. Consider $n>3$ and define $A(n)= 1!+2!+\cdots+n!$

It seems that if $p$ is the largest prime divisor of $A(n)$, then the $p$-adic valuation of $A(n)$ is 1. I’ve tested several hundred cases on mathematica. This function is a translate of Kurepa’s “left factorial” function, but I haven’t found anything in the literature related to this specific question. Perhaps this has an elementary proof.

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    $\begingroup$ The $A(n)$ are tabulated at oeis.org/A007489 and there are links to tables of factorizations into primes. $\endgroup$
    – Gerry Myerson
    Commented Aug 11 at 3:50
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    $\begingroup$ The valuation of the largest prime divisor is 1 with probability of approximately $1 - \frac{1}{\sqrt n}$, right? The sum of $\frac1{\sqrt{n!}$ from $n=100$ is approximately $10^{-79}$, so heuristically this seems very likely (of course, these aren't random numbers) $\endgroup$
    – Daniel Weber
    Commented Aug 11 at 11:55

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