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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. ?

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    Please edit your question so it asks what you actually want to ask.2012-07-10

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