Approximations for the number of divisors of an integer

Asked Jul 09 '12 at 11:25

Active Jul 09 '12 at 12:45

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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. ?

edited Jul 09 '12 at 12:45

asked Jul 09 '12 at 11:25

o17t H1H' S'k

2

For (a),see http://mathworld.wolfram.com/DistinctPrimeFactors.html. For (b), see http://en.wikipedia.org/wiki/Divisor_function#Approximate_growth_rate – lhf Jul 09 '12 at 11:49
so are my suggested answers correct now? – o17t H1H' S'k Jul 09 '12 at 12:12
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Perhaps you could add an answer with the details or some indication of how you have found these answers. – lhf Jul 09 '12 at 12:14
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There are about $x/\log x$ primes up to $x$, and their product is about $e^x$. Letting $n$ be that product, the number of prime factors is thus about $\log n/\log\log n$, which is not $O(\log\log n)$. – Gerry Myerson Jul 09 '12 at 12:33
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Maybe I'm missing something, but I don't see how a number can have "non-distinct divisors"... – J. M. ain't a mathematician Jul 09 '12 at 12:51
@gerry, i think you are looking at the primorial case which is maximal, while i am looking at the average case – o17t H1H' S'k Jul 09 '12 at 12:51
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@eyaler: if you want a big O bound you should look at the maximal examples. – tomasz Jul 09 '12 at 12:52
Please edit your question so it asks what you actually want to ask. – Gerry Myerson Jul 10 '12 at 01:36

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