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With the question I made about primes, I noticed people enjoy the subject, so here's another thought: let k be a positive integer; how many primes are there from 1 to k?

There's probably no exact expression for (let's call it) N(k) — and, unlike the other question, asking for $\lim_{k \to \infty} N(k)$ is silly because it's obviously infinity —, so, in a lenient variant of the question, does N(k) asymptotically reach a certain function of k as it becomes larger?

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@Bill Dubuque That means the function I called $N$ is called $\pi$, and $\pi(k) \sim\! k / ln \space k$. –  Luke May 11 '11 at 1:07

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up vote 5 down vote accepted

This is the well-known and well-studied prime-counting function, which you can read plenty about on its Wikipedia article.

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Look up the Prime Number Theorem.

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