Is there a formula that can tell us how many distinct prime factors a number has? We have closed form solutions for the number of factors a number has and the sum of those factors but not the number of distinct prime factors.
So for example: \begin{array}{rr} \text{number} & \text{distinct}\atop \text{prime factors} \\ 1&0 \\ 2&1 \\ 3&1 \\ 4&1 \\ 5&1 \\ 6&2 \\ 7&1 \\ 8&1 \\ 9&1 \\ 10&2 \end{array}
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$\begingroup$This is a historically interesting question as it led Hardy and Ramanujan to lay the foundation to probabilistic number theory in course of their solution to this problem. Given $n$ there is no non-trivial deterministic closed form formula for the number of distinct prime factors of $n$. However we have very good probabilistic formula for the same.
Hardy and Ramanujan proved that for almost all integers, the number is distinct primes dividing a number $n$ is formula
$$ \omega(n) \sim \log\log n. $$
We can do much better than the Hardy-Ramanujan estimate and find and estimate of $\omega(n)$ which can be bounded by normal distribution. Erdos and Kac imporved the estimate of $\omega(n)$ and proved that
$$ \lim_{x \to \infty} \frac{1}{x}\# \Big\{n\le x, \frac{\omega(n) - \log\log n}{\sqrt{\log\log n}} \le t \Big\} = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{t}e^{-\frac{u^2}{2}}du $$
This formula says that if $n$ is a large number, we can estimate the distribution of the number of prime factors for numbers of this range. For example we can show that around 12.6% of 10,000 digit numbers are constructed from 10 distinct prime numbers and around 68% (±$\sigma$) are constructed from between 7 and 13 distinct primes.
$\endgroup$ 6 $\begingroup$This is commonly known as $\omega(n)$. See Wikipedia.
$\endgroup$ $\begingroup$We do not have the closed form solutions you claim. We have formulas that assume that we know the prime factorization.
$\endgroup$ $\begingroup$If you want to calculate $\omega(n)$ for a large range of $n$, it's simple to modify the Sieve of Eratosthenes to count distinct prime factors. Instead of marking multiples of each prime, keep a counter for every number and increment instead of marking. SO code example
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