# Question regarding $C(n)$ and $B(n)$ in Hanson's proof that $\prod\limits_{p^a \le n} p < 3^n$

In Denis Hanson's proof, he defines two terms:

(1) $$B(n)$$: which is the Least Common Multiple of $$\{1,2,\dots,n\}$$

$$B(n) = \prod\limits_{p^a \le n}p$$

(2) $$C(n)$$: an integer

$$C(n) = \dfrac{n!}{\lfloor n/a_1 \rfloor\lfloor n/a_2 \rfloor\lfloor n/a_3 \rfloor\dots}$$

where $$a_1 = 2$$ and $$a_{n+1} = a_1 a_2 \dots a_n + 1$$

The proof ends with the following conclusion:

$$C(n) < 3^n$$

I am completely lost where $$C(n)$$ was shown to be equal or greater to $$B(n)$$.

Am I misunderstanding?

If someone could explain how $$C(n) \ge \text{LCM}(1,2,\dots,n)$$, I would greatly appreciate it.

Since $$\alpha_p$$ is such that $$p^{\alpha_p}$$ is the highest power of $$p$$ not exceeding $$n$$, we can write $$\alpha_p=[\log_pn]$$ So, $$B(n)$$ is written as $$B(n)=\prod_{p\le n}p^{[\log_pn]}$$
From LEMMA 2., we get $$[\log_pn]\le \beta_p(n)$$ from which $$B(n)=\prod_{p\le n}p^{[\log_pn]}\le \prod_{p\le n}p^{\beta_p(n)}=C(n)$$ follows.