HMMT 2022 A10

Compute the smallest positive integer $n$ for which there are at least two odd primes $p$ such that

$$\sum_{k=1}^{n}(-1)^{\nu_{p}(k!)}<0$$

Note: for a prime $p$ and a positive integer $m, \nu_{p}(m)$ is the exponent of the largest power of $p$ that divides $m$; for example, $\nu_{3}(18)=2$.