It is well known and easy to prove that the smallest prime factor of an integer $n$ is at most equal to $\sqrt n$. What can be said about the largest prime factor of $n$, denoted by $P_1(n)$? In particular: What is the probability that $P_1(n)>\sqrt n$ ? More generally, what is the expected val...

Asymptotically, only the primes matter. If $p$ is the smallest prime factor of a composite $k$, then $k \geqslant p^2$, and so the contribution of composites to the average can be estimated from above by $\sqrt{n}$ via $$\sum_{p \leqslant \sqrt{x}} \bigg\lfloor\frac{n}{p}\biggr\rfloor\cdot p \le...

Explanation: Last year in math class, on homework we would occasionally get these extremely simple, although equally annoying questions called diamond puzzles. These were basically questions where we would be given a sum, and a product then were asked to find the two numbers which when multiplie...