[seqfan] Division with no visible digits

[Eric Angelini]

Hello SeqFans,
I've had this (silly) idea: what about a sequence where
the result of a(n) divided by a(n+1) has no digit in
common with a(n) or a(n+1)?

S = 1,2,3,4,5,6,8,11,15,25,22,24,27,9,12,16,32,...

S was always extended with the smallest available integer
not yet in S and not leading to a contradiction.

Example:
1/2 = 0.5 
2/3 = 0.666666666666666...
3/4 = 0.75
4/5 = 0.8
5/6 = 0.833333333333333...
... but no 6/7 here because 6/7 = 0,857142... with a "7"
6/8 = 0.75
... no 8/7 here as 8/7 = 1.142857... with both "7" and "8"
... no 8/9 as 8/9 = 0.88888888888... with lots of 8's
... no 8/10 here, of course (8/10 = 0.8)
8/11 = 0.72727272727272...
The next term will be 15 as 11/15 = 0.73333333333...
etc.

Question:
S is finite as the next term will be 20 -- with no possible
successor.
Is there an infinite such sequence if one doesn't start with
a(2)=2?

Best,
É.