Sequence Without Sevens
Java, 122 121 98 97 bytes

int c(int n){int a=0,b=0,c=n;for(;n--&gt;0;c+=a=((--b+&quot;&quot;).indexOf(55)&amp;b%7)&gt;&gt;31);return a&lt;0?-b:c(c);}

-23 bytes thanks to @Nevay.
-1 byte thanks to @Neil.
Explanation:
Try it here.
int c(int n){                            // Method with integer as parameter and return-type
  int a=0,                               //  Flag integer
      b=0,                               //  Regular counter
      c=n;                               //  7-counter
  for(;n--&gt;0;                            //  Loop over the input
    c+=                                  //   Append `c` with:
       a=                                //    Set `a` to:
         ((--b+&quot;&quot;).indexOf(55)&amp;b&amp;7)      //     If `b` is divisible by or contains 7:
                                         //     (and decrease `b` by 1 in the process)
                                   &gt;&gt;31  //      Bitwise right shift so `a` is either `-1` or `0`
  );                                     //  End of loop
  return a&lt;0?                            //  If `a` is now -1:
    -b                                   //   Return the regular counter `b` (as positive)
   :                                     //  Else:
    c(c);                                //   Recursive call with 7-counter `c` as input
}                                        // End of method