# Determine the number of factors for extremely large numbers.

An offshoot from a related question, is there a way to determine the number of possible factors (odd, even, prime, etc.) for extremely large integers without actually factoring them?

Even an estimation would help as long as it has some relevance to the number in question.

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A very crude upper bound for the number of factors of $n$ is$\lfloor \ln_2(n) \rfloor$ (here $\lfloor . \rfloor$ denotes the floor function) : since $n = 2^{\ln_2(n)}$ and each factor of $n$ is $\ge 2$, there can be no more than $\lfloor \ln_2(n) \rfloor$ factors. The only lower bound that works for every number is $1$ (if $n$ is prime, this bound is sharp). – Joel Cohen Jul 29 '12 at 20:50
Note that in my previous comment, I took "factor" to mean "prime factor" and not "divisor". – Joel Cohen Jul 29 '12 at 20:59
@JoelCohen: I do not want to discriminate. I'm looking for all possible integer divisors (odd, even, prime, etc.). – Raheel Khan Jul 29 '12 at 21:07
see mathoverflow.net/questions/43103/… Meanwhile if you email me, you can have a pdf of the 1988 survey by J.-L. Nicolas with these results and explanations. – Will Jagy Jul 29 '12 at 21:11
@WillJagy: Where should I email you? – Raheel Khan Jul 30 '12 at 16:54

If the prime factorization of $N$ is $p_1^{d_1} \ldots, p_n^{d_n}$, then $N$ has $(d_1+1)\ldots(d_n+1)$ divisors. This number of divisors is denoted $d(N)$. A crude upper bound on $d(n)$ is $2 \sqrt{ n}$. A better estimate is
$\log d(n) \le \dfrac{\log n}{\log \log n} (\log 2 + O(1/\log\log n))$