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Can anyone suggest an algorithm faster than $\Theta(n^{2})$ for computing the following function:

$$||n||:=\frac{1}{\max\{k \in \mathbb{N}: 1|n, 2|n,\ldots,k|n\}}$$

Raphael
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Jackson Walters
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  • Why is it $O(n^2)$? Don't you just need to do for k = 1 to infinity, if n mod k != 0, return k-1? I wonder if I misunderstood your question. – Karolis Juodelė Mar 25 '13 at 06:47
  • What have you tried? Also, "faster than $O(n^2)$" makes no sense. – Raphael Mar 25 '13 at 10:32
  • If the weird notation means select the largest $k$ that divides $n$, it is just $n$. $O(1)$. – vonbrand Mar 25 '13 at 11:10
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    Of course $O(n^2)$ means $n^2$ is an upper bound, but I think you knew what I meant. @vonbrand The "weird" notation is standard and that is not what that means. It asks for the largest $k$ such that $i | n$ for all $1 \le i \le k$. For instance, $||n||=1$ for $n$ odd. – Jackson Walters Mar 25 '13 at 13:27

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Store a list of all "factorials" (least common multiplies of $\{1,\ldots,k\}$) and use binary search. That reduces the number of divisions in the worst case. If the numbers $n$ you are getting are "random" then you might want to start with smaller trial divisions, or perhaps with an exponentially growing $k$, i.e. try $1,2,4,8,\ldots$ until you find some $k$ for which the LCM does not divide $n$, then take it from there using binary search.

By the way, since the LCM grows exponentially, even your algorithm only requires $\log n$ divisions. Each division takes time $\tilde{O}(n)$, so in total both your algorithm and mine are $\tilde{O}(n)$.

Yuval Filmus
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