Seven plus

The Seven plus operation transforms a digit 7 into a plus sign.

The integer 170 becomes 1+0 which is 1.
We want that a(n), transformed by the Seven plus operation, divides a(n+1).
The term a(n+1) has to contain at least one digit 7 in a "good" position (see below).
No two identical terms in the sequence S – which should be the lexico-first of its kind.

For a(1) = 170, we get:

S = 170, 171, 172, 174, 175, 270, 176, 273, 275, 371, 272, 276, 376, 279, 374, 378, 473, 476, 370,...

We see indeed that:
170 = 1+0 = 1 divides 171;
171 = 1+1 = 2 divides 172;
172 = 1+2 = 3 divides 174;
174 = 1+4 = 5 divides 175;
175 = 1+5 = 6 divides 270; etc.

Numbers we don't want to see in S:
— no term starting with a 7 (ex. 754 or 7574);
— no term ending with a 7 (ex. 127 or 1727);
— no term with 2 or more consecutive 7 (ex. 1778);
— between two 7, no string with a leading 0 (ex. 170578 – but 17078 is ok as this term would produce 1+0+8 = 9).

Where does S (S for "seven") go? 
As usual, please forgive my hand mistakes.
Best,
É.