Just Simplify That Power Tower!

Number Theory Level 1

$\large \text{lcm} (2^1 ,2^2 ,2^3,\ldots, 2^n) = 2^{2^{2^2}}$

What is the value of $$n$$ satisfying the equation above? 

Notation: $$\text{lcm}(\cdot)$$ denotes the lowest common multiple function.

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