Prime numbers are essential in various areas of cybersecurity, from encrypted messages to secure payments, HTTPS websites, and RSA cryptography techniques.Numerous computational and theoretical studies on these "unbreakable" numbers in arithmetic ((+)) are regularly published. Here we continue the article [15] by considering sequences of consecutive primes of the type considered then, primes with a certain initial digit, primes without a certain digit, primes with a certain digit, primes with a certain last digit, primes with a certain digit in a certain position, etc. We will calculate the maximum number of consecutive elements in such sequences, as well as the probabilities of obtaining a certain consecutive number from one of these primes. We will also seek to determine whether there is a possible order for the maximum lengths of sequences of such primes with reference to these various digits, and in which cases this may be a Benford-type order (inverse natural order of integers).We will conclude with some results concerning the maximum differences between any consecutive primes or those possessing a certain property.