Pandigital primes

A base $\scriptstyle b\,$ pandigital prime $\scriptstyle p\,$ is a prime number that is also a pandigital number in the given base. The smallest base $\scriptstyle b\,$ pandigital prime usually starts with the string "1012" rather than "102" for the smallest pandigital composite number, after which follows a concatenation of the other base $\scriptstyle b\,$ digits more or less in order (since having exactly one instance of each digit probably means the number is divisible by $\scriptstyle b-1\,$, at least when $\scriptstyle b\,$ is even.)

Sequences of pandigital primes base $\scriptstyle b\,$

$b\,$	A-number	Sequence
2	A138837	{2, 5, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, ...}
3		Have not looked yet
4		Have not looked yet
5	A175277	{3319, 3323, 3347, 3469, 3491, 3539, 3547, 3559, 3571, 3607, 3613, 3617, 3691, 3823, 3847, 3863, 4019, ...}
6		Have not looked yet
7		Have not looked yet
8	A175271	{17119607, 17120573, 17121077, 17127839, 17128931, 17132347, 17135413, ...}
9		Have not looked yet
10	A050288	{10123457689, 10123465789, 10123465897, 10123485679, 10123485769, ...}

The base doesn't have to get too large for pandigital primes to get quite large. By most standards, 36 is not a terribly large number, yet the smallest pandigital prime in base 36 is already larger than the largest known Mersenne prime prior to 1952. I have searched in vain for a compact way to express these numbers in terms of addition of powers of smaller numbers. However, the nature of these numbers provides a very efficient shortcut: for example, the smallest base 36 pandigital prime is 10123...STUXYZWV (the ellipsis omitting the predictable digits 4, 5, 6, 7, etc., up to P, Q, R).