Question 74
Compute the smallest positive integer $n$ for which there are at least two odd primes $p$ such that
$$
\sum_{k=1}^{n}(-1)^{\nu_{p}(k!)}\lt 0
$$
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, $\nu_{3}(18)=2$.
Proposed by: Krit Boonsiriseth
$$
\sum_{k=1}^{n}(-1)^{\nu_{p}(k!)}\lt 0
$$
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, $\nu_{3}(18)=2$.
Proposed by: Krit Boonsiriseth