Trường THPT Chuyên Bắc Ninh

$${{v}_{p}}(n!)=\sum\limits_{i=1}^{\infty }{\left\lfloor \frac{n}{{{p}^{i}}} \right\rfloor }=\sum\limits_{i=1}^{\left\lfloor {{\log }_{p}}n \right\rfloor }{\left\lfloor \frac{n}{{{p}^{i}}} \right\rfloor } \sim \frac{n}{p-1}$$ $$\sum\limits_{p \text{ nguyên tố}}^{p\le n}{\frac{1}{p}} \sim \log\log n $$ $$\prod\limits_{p\le n}{p}\le {{4}^{n}}$$ $$\lim \frac{{{\left( \pi (n) \right)}^{2}}}{n}=+\infty $$

TNV VNOI Gen 3