Is there a function $s(n)$ which tells the amount of factors any $n$ has? And if not, is it possible to write one?
Is there a function $s(n)$ which tells the amount of factors any $n$ has?
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elementary-number-theory
prime-numbers
terminology
arithmetic-functions
divisor-counting-function
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0Is [*this function*](https://en.wikipedia.org/wiki/Divisor_function) what you are looking for? – 2017-01-15
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0Yes there is. You just defined it. – 2017-01-15
1 Answers
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It is called the Divisor (Counting) Function, usually denoted $\tau(n)$, $\sigma_0(n)$ or (less fancy) $d(n)$.
If $n=\prod_{k=1}^rp_k^{a_k}$ is the prime factorisation of $n$, we have $$\tau(n)=\prod_{k=1}^r(a_k+1).$$
For questions about it, see divisor-counting-function.
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0Thanks for this clear and direct answer. Could you please explicitly define $a_k$ and $r$ for me? I'm just a little confused on what those values represent. – 2017-01-15
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1$p_k$ are the prime numbers dividing $n$, $r$ is the number of distinct primes that divide $n$, $a_k$ is the largest integer for which $p_k^{a_k}$ divides $n$. https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic https://proofwiki.org/wiki/Definition:Prime_Decomposition but I don't think I should write about prime factorization here, so I'm leaving the answer as is. – 2017-01-16