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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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On the $p$-adic valuation of the sum of the first $n$ factorials

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 …