Htykuut

I was playing around with the möbius inversion formula and factorials and I came up with an equation that I found interesting:

$$prod_{ige2}leftlfloorfrac{x}{i}rightrfloor! = prod_{i ge 2}prod_{j ge 1}left(leftlfloorfrac{x}{ij}rightrfloor!right)^{-mu(i)}$$

I'm not sure if this equation is valid.

Here is the reasoning:

(1) Let $text{lcm}(x) = $ the least common multiple of ${1, 2, 3, dots, x}$

(2) Using a well known equation:

$$x! = prod_{i ge 1}text{lcm}left(leftlfloorfrac{x}{i}rightrfloorright)$$

(3) Using mobius inversion, I believe that this is now valid:

$$text{lcm}(x) = prod_{i ge 1}left(leftlfloorfrac{x}{i}rightrfloor!right)^{mu(i)}$$

(4) Combining the two equations gets me to:

$$x! = prod_{j ge 1}prod_{i ge 1}left(leftlfloorfrac{x}{ij}rightrfloor!right)^{mu(i)}$$

(5) Since order of the products doesn't matter:

$$x! = prod_{i ge 1}prod_{j ge 1}left(leftlfloorfrac{x}{ij}rightrfloor!right)^{mu(i)}$$

(6) Since for $i=1,j=1$, $x! = left(frac{x}{ij}!right)^{mu(i)}$:

$$1 = prod_{i ge 1}prod_{j ge 1text{ and }jne1text{ if }i=1 }left(leftlfloorfrac{x}{ij}rightrfloor!right)^{mu(i)}$$

(7) We can now use division to separate the remaining cases of $i=1$ so that we get:

$$prod_{j ge 2}leftlfloorfrac{x}{j}rightrfloor! = prod_{i ge 2}prod_{j ge 1}left(leftlfloorfrac{x}{ij}rightrfloor!right)^{-mu(i)}$$

Is this equation correct? And if correct, is it well known?