For fixed positive integers a and c, the prime sequences of the form ap_n + cb, where p_n is the nth prime and gcd(a,cb) = 1, can share terms as b varies. When the sequences share terms, we say that they overlap. Furthermore, when the sequences overlap with each other or another common prime sequence of a similar form, we say that these prime sequences are in the same family. We show that when a and cb have opposite parity, the number of families of prime sequences is Φ(a), where Φ(a) is Euler’s totient function. In other words, the number of families depends only on a. Several of these overlapping prime sequences, their families, and the pseudocode used to generate the sequences will be included in the presentation.