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Let $r$ be a uniformly random integer between $1$ and $N$, for some large enough $N$ (i.e., I'm only interested in the asymptotics).

  • What is the expected largest prime factor of $r$? Is there a good upper bound on this?

  • Let $t < N$ be such that $\Pr\Big[r $ has no prime factor > $t\Big] = \Omega(1/\log^c(N))$ for some constant $c > 0$. How small can we make $t$? Is $t = \operatorname{Li}_2(N)$ achievable?

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