[seqfan] Strict divisors and digits of a(n)

[Eric Angelini]

Hello SeqFans,
I don't know if "strict divisor" (SD) is
the English translation of the French
"strict diviseur" -- anyway you will
understand if I say that the SDs of 
6 are 1,2 and 3, and (second example),
the SDs of 28 are 1,2,4,7 and 14.

Now consider S:
 a(n) will _not_ be part of S if one digit 
of a(n) can be found in one of its SDs:

S=2,3,4,5,6,7,8,9,23,27,29,34,38,...

SDs of the above integers (2nd column):
2=1
3=1
4=1,2
5=1
6=1,2,3
7=1
8=1,2,4
9=1,3
23=1
27=1,3,9
29=1
34=1,2,17
38=1,2,19
...
We see that for the above integers,
no digit at the left of the equal sign
is repeated at the right of the sign.

My (naive?) question is:
-- Is S finite?

Best,
É.