This large number(24 digits) is a prime number and is called, as you know, a "Truncatable Prime":
357,686,312,646,216,567,629,137
It has a very special and unique property which is this: If you start from the left and remove the first 3, the remaining number is STILL a prime number! If you remove the next number 5, the remaining number is still a prime number! If you continue and remove the next number 7, the remaining number is still a prime number. And if you continue with this process, the remaining number continues to be a prime number, all the way to the last digit of 7!!!.
Isn't that remarkable?? Your sequence of primes is something like that, but much less impressive!!!.