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$0,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,\cdots$

Wolfram alpha says that it is the number of distinct primes in n. Does it have another closed form? (This is from an introductory book on fourier series)

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    If you drop the initial $0$, you can find more possibilities in the [OEIS](http://oeis.org/search?q=1%2C1%2C1%2C1%2C2%2C1%2C1%2C1%2C2%2C1%2C2%2C1%2C2%2C2%2C1%2C1%2C2%2C1%2C2%2C2%2C2%2C1%2C2%2C1%2C2%2C1%2C2%2C1) meaning that the next term can plausibly be $3$, $4$ or $5$2011-04-18

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Depends a bit on what you mean by closed form. It has a standard notation, $\omega(n)$, as you can see at the reference Theo gives. But if by a closed form you mean a way to calculate it that's easier than factoring $n$ and counting the number of primes you get, that would have to rate as exceedingly unlikely.

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Some time ago I found the following formula $\large \omega(n)=\sum_{p\in\mathbb{P}}\;\sum_{r=1}^{\infty}{\frac{1}{p^{r+1}}}\sum_{j=1}^{p-1}{\;\;\;\sum_{k=0}^{\;p^{r+1}-1}\cos\left\lbrace 2k\pi \left(\frac{n+(p-j)p^{r}}{p^{r+1}}\right)\right\rbrace}$

Where $\mathbb{P}$ is the set of prime numbers. A closely related one is

$\large \Omega(n)=\sum_{p\in\mathbb{P}}\;\sum_{r=1}^{\infty}{\frac{r}{p^{r+1}}}\sum_{j=1}^{p-1}{\;\;\;\sum_{k=0}^{\;p^{r+1}-1}\cos\left\lbrace 2k\pi \left(\frac{n+(p-j)p^{r}}{p^{r+1}}\right)\right\rbrace}$

They are useless... just factorize $n$ or... convert the to something useful!