[seqfan] "What primes P have P^P containing the string 'P' as substring'?"

[Jonathan Post]

JM Bergot asked me in an email: "What primes P have P^P containing the
string 'P' as substring'?"

5^5 = 3125 has "5" as substring.

11^11 = 285311670611 has two substrings of "11"

17^17 = 827240261886336764177  has "17" as 3rd and 2nd digit from right-hand.

19^19 = 1978419655660313589123979 has a "19" as its left end, and another later.

31^31 = 17069174130723235958610643029059314756044734431

has a "31" at right-hand end, and a second to the left a ways.

A051674 (n-th prime)^(n-th prime).

Is there a sequence in there, base and prime related, struggling to be
interesting to anyone on seqfans?

Eventually, all P^P are pandigital.  But not "normal." Right?

Best,

Jonathan Vos Post