Just Simplify That Power Tower!

lcm(21,22,23,,2n)=2222 \large \text{lcm} (2^1 ,2^2 ,2^3,\ldots, 2^n) = 2^{2^{2^2}}

What is the value of nn satisfying the equation above?

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

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