Does any number, p, have a number of prime factors greater than ln(p)?

Question

I've wrote a very silly algorithm for prime decomposition, but I can't be certain that it works unless I can find a bound $K(p)$ such that for any number, $p$, the number of prime factors of $p$ is less than $K(p)$ (counting multiplicity, i.e. $4=2*2$ has two factors). Is $\ln(p)$ such a bound?

Answer 1

If you count according to multiplicity, $2^n$ has $n$ prime factors while $\ln (2^n)=n \ln 2 \approx 0.693 n$, so it fails. To be safe you could use $\frac 32 \ln p$. As $2$ is the only prime less than $e$ you need a lot of factors of $2$ to make $\ln p$ fail.