Suppose we are comparing products of powers of primes p vs. n!:
p₁^(2n/p₁)∙p₂^(2n/p₂)∙p₃^(2n/p₃)∙p₄^(2n/p₄)∙... vs. n!
If p₁ = 2, and p₂ = 3, then the right-hand-side (RHS) > left-hand-side (LHS) above a certain n
If p₁ = 2, p₂ = 3, and 5 ≤ p₃ ≤ 17, then the LHS > RHS for all n
But for p₁ = 2, p₂ = 3, and p₃ ≥ 19, then again the RHS > LHS above a certain n
Right now I'm using Wolfram Alpha to investigate these combinations.
Wondering if there is a deterministic way to decide if these products of powers of primes is asymptotically less than or greater than n!?
If you are using all the primes not greater than $n$ then $p_1^{\frac{2n}{p_1}}\cdots p_k^{\frac{2n}{p_k}}$ should be greater than $n!$.
We know that $n!$ is equal to $\prod p_i^{[\frac{n}{p_i}]+[\frac{n}{p_i^2}]+...}$
($[x]$ denotes the integer part of $x$). So we only have to compare the exponents.
The exponent of every prime in the factorial is not greater than $\frac{n}{p_i-1}$.
But $\frac{n}{p_i-1}\leq \frac{2n}{p_i}$ for every $p\geq 2$
I do not understand why you find that RHS > LHS