Mathematica, 53 bytes

(For[i=n=#,n>1,--n,If[n∣i,i+=n(2Mod[i/n,2]-1)]];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 nth Shotgun number.