# Mathematica, <s>53</s> 49 bytes

    (For[i=n=#,n>1,--n,If[n∣i,i+=Mod[i,2n]2-n]];i)&

I decided to golf my reference implementation. The `∣` is the Unicode symbol for "divides", and counts for 3 bytes. Otherwise, this uses the same algorithm as everyone else.

It defines an unnamed function which takes `n` as a single parameter and returns the `n`th Shotgun number.