Is there a theorem, lemma, or proof somewhere that proves an upper bound for the number of factors that a number can have?
If not, would it be fairly trivial to prove that it is $log_2 n$?
Is there a theorem, lemma, or proof somewhere that proves an upper bound for the number of factors that a number can have?
If not, would it be fairly trivial to prove that it is $log_2 n$?
Let ${d(n)}$ denote the number of factors of $n$.
There is a useful bound which has applications in many area:
${d(n)\le n^{O(1/\log \log n)} = \exp(O(\frac{\log n}{\log\log n}))}$.
Maybe Tao's blog can help you understand the answer better.
https://terrytao.wordpress.com/2008/09/23/the-divisor-bound/
given that log2(24) is approx. 4.5, but 24 has 8 factors. I have to say, that this is not an upper bound.
I personal looked for an upper bound recently to the number of factors function and I got sqrt(3*x), it might not be the tightest upper bound. but it works.
look at the image bellow