666 and godly numbers

[Max Alekseyev]

SeqFan,

The following question was asked in fido7.ru.math newsgroup:

A positive integer n is called "godly" if for every integer base b>6,
the base-b representation of n contains no substring "666" (i.e., no
three consecutive base-b digits equal 6).
Is there an infinite number of godly numbers?

Currently I have no proof for that (but I believe it's not a hard problem).

The godly numbers are not in OEIS. But do you know if there are any
similar sequences?

Regards,
Max