Just to clarify, in this case p = 77,232,917 is prime, but for Mersenne primes in general p is just a positive integer.

Not quite. If p isn't prime then 2^p - 1 isn't prime.

Refinable functions, used for constructing wavelets and subdivision schemes, seem similar: https://en.wikipedia.org/wiki/Refinable_function and https://arxiv.org/abs/1012.2453