In Denis Hanson's proof that $\prod\limits_{p^a \le n} < 3^n$, I am confused by the final step.
He proceeds to show that:
$$C(n) = \frac{n!}{\lfloor n/a_1\rfloor!\lfloor n/a_2\rfloor!\lfloor n/a_3\rfloor!\dots} < e^{k-3/2}n^{k-3/2}w^n$$
for $k > 2$ and $w < 2.952$
Then, he mentions that:
A check of tables reveals that $C(n) < 3^n$ for $n > 1300$
What table is he referring to? And how is this possible given the value of $w$. How can $(en)^{k-3/2}$ possibly be less than $3^{n - 2.952}$?
Am I misunderstanding the last step? Did I mistake a mistake in any of my statement above? If not, could someone send me a reference to the tables and explain how the tables establish the conclusion.