No string of digits divisible by k

[Eric Angelini]

Hello SeqFans,
could someone examine this sequence:

- Zeroless integers having no string of digits divisible by 3.

I think it starts like this:

1,2,4,5,7,8,11,14,17,22,25,28,41,44,47,52,... (not OEIS)

... but I'm quite sure the sequence is finite. Example:

715 will not fit because "15" is a substring divisible by 3;
716 neither because "6" is divisible by 3,
717 neither because "717" is divisible by 3;
718 neither because of "18";
719 neither because of "9";
720 neither (has a "0")
etc.

If we replace the definition by:

- Zeroless integers having no string of digits divisible by k.

... what could be the smallest k leading to an infinite sequence
-- if such k exists -- or to the longest finite sequence?

Sorry if this is hold hat.

Best,
É.