Puzzle 832. Follow-up to Puzzle 828.

In our puzzle 828 we asked for the smallest sequence of distinct primes such that any two consecutive members, p & q satisfy the following three conditions:

Neither p & q, p+q nor p*q contains the digits "2", "0", "1" or "6".

In puzzle 828 we soon discovered that the asked sequence is composed of primes ending in the digit "7".

Here we ask if are there primes p ending in "7" such that q is impossible.

Q. Find the smallest five of such of these p primes ending in "7", if they exist.