0
$\begingroup$

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. ?

  • 2
    For (a),see http://mathworld.wolfram.com/DistinctPrimeFactors.html. For (b), see http://en.wikipedia.org/wiki/Divisor_function#Approximate_growth_rate2012-07-09
  • 0
    so are my suggested answers correct now?2012-07-09
  • 1
    Perhaps you could add an answer with the details or some indication of how you have found these answers.2012-07-09
  • 1
    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 $O(\log\log n)$.2012-07-09
  • 3
    Maybe I'm missing something, but I don't see how a number can have "non-distinct divisors"...2012-07-09
  • 0
    @gerry, i think you are looking at the primorial case which is maximal, while i am looking at the average case2012-07-09
  • 1
    @eyaler: if you want a big O bound you should look at the maximal examples.2012-07-09
  • 0
    Please edit your question so it asks what you actually want to ask.2012-07-10

0 Answers 0