[seqfan] a(n), a(n+1), [a(n)+a(n+1)] share no digits

[Eric Angelini]

Hello SeqFans,
we consider two consecutive integers ("a" and "b") from a
monotonically increasing seq S, and their sum "c"; we want
that:

"a" shares no digit with "b"
"b" shares no digit with "c"
"c" shares no digit with "a"

Starting with 1, do we have:

S = 1,2,3,4,5,6,7,8,9,11,22,33,42,51,249,302,449,... ?
(to prolong S, take the smallest integer not leading to a
 contradiction)

Does S stop at some point?
(is this old hat, BTW?)
Best,
É.