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Help me with this question
By shrohit_007, history, 14 months ago, In English

the question is very simple we just need to calculate total number of numbers which have exactly 4 divisors for ex 6, 8, 10 these are all of the forms p^3 or p*q but here n<=10^11

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14 months ago, # |
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14 months ago, # |
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Here $$$\pi(n)$$$ denotes prime counting function, i.e number of primes not greater than $$$n$$$.

Numbers form $$$p^3$$$ can be easily counted, it's just $$$\pi \left(\lfloor \sqrt[3]{n} \rfloor\right)$$$.

To calculate numbers form $$$pq$$$ recall, that

$$$ \sum\limits_{k = 1, k = pq}^n 1 = \sum\limits_{p \leqslant n} \sum\limits_{p < q \leqslant \frac{n}{p}} 1 = \sum\limits_{p \leqslant \sqrt{n}} \pi \left( \lfloor \frac{n}{p} \rfloor \right) - \pi(p)$$$

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Primes and $$$\pi$$$ up to $$$\sqrt{n}$$$ can be calculated straightforward using Eratosphenes sieve.

Also you can calculate values of $$$\pi\left( \lfloor \frac{n}{k} \rfloor \right)$$$ for all $$$k \geqslant 1$$$ using $$$O(n^{2/3})$$$ time as described here https://mirror.codeforces.com/blog/entry/91632

IMHO this question is not very simple as it requires some knowledge.

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