Given an integer $n$, I want to know the asymptotic order of:
a. the number of distinct prime factors
b. the number of non-distinct prime factors
c. the number of distinct divisors
d. the number of non-distinct divisors
To my understanding, the references in the comment below suggest the following:
a. $\mathcal{O}(\log\log n)$
b. $\mathcal{O}(\log\log n)$
c. $\mathcal{O}(n^{\frac{1}{\log \log n}})$
d. ?