Divisibility Streak

Java 8, 44 42 41 39 bytes

^{Crossed out 44 is still regular 44 ;(}

n->{int r=0;for(;~-n%--r<1;);return~r;}

-2 bytes thanks to @LeakyNun.
-1 byte thanks to @TheLethalCoder.
-2 bytes thanks to @Nevay.

Explanation:

Try it here.

n->{                 // Method with integer as parameter and return-type
  int r=0;           //  Result-integer (starting at 0)
  for(;~-n%--r<1;);  //  Loop as long as `n-1` is divisible by `r-1`
                     //   (after we've first decreased `r` by 1 every iteration)
  return~r;          //  Return `-r-1` as result integer
}                    // End of method

Haskell, 35 bytes

f n=[k|k<-[1..],rem(n+k)(k+1)>0]!!0

Try it online!

Using until is also 35 bytes

f n=until(\k->rem(n+k)(k+1)>0)(+1)1

Pyth, 6 5 bytes

f%tQh

Try it online!