Charcoal, 11 bytes

≔…·²ＮθＩ⁺θΠθ

Try it online! Link is to verbose version of code.

≔…·²Ｎθ

Create a list from 2 up to n inclusive.

Ｉ⁺θΠθ

Vectorised add the list to its product.

Of course, the given formula is just an existence proof and there are usually lower runs. For instance, the LCM is readily proven to produce a run:

Ｎθ≔…·²θηＷ⌈﹪±θη≧⁺ιθＩ⁺θη

Try it online! Link is to verbose version of code. Or more efficiently:

Ｎθ≔…·²θηＦη≔⌊Φ×⊖ηθ¬﹪κιθＩ⁺θη

Try it online! Link is to verbose version of code.