33 bytes for a more efficient version that can suffer from floating-point inaccuracy:

ＮθＩＥ⟦¹±¹⟧×ι⌈Ｅ₂θ×ιＸ×ι⌊×ιＸθ∕¹⁺²λ⁺²λ 

Try it online! Link is to verbose version of code. Explanation: Port of @bsolech's golf to @corvus_192's Python answer.

Ｎθ Input `N` as a number ¹ Literal integer `1` ¹ Literal integer `1` ± Negated ⟦ ⟧ Make into list Ｅ Map over units θ Input `N` ₂ Square root Ｅ Map over implicit range θ Input `N` Ｘ Raised to power ¹ Literal integer `1` ∕ Divided by λ Current value ⁺ Plus ² Literal integer `2` × Multiplied by ι Current unit ⌊ Floored to integer × Multiplied by ι Current unit Ｘ Raised to power λ Current value ⁺ Plus ² Literal integer `2` × Multiplied by ι Current unit ⌈ Take the maximum × Multiplied by ι Current unit Ｉ Cast to string Implicitly print