Modified Euler's Totient function for counting constellations in reduced residue systems

1: Your formula for $\phi(-k)(n)$ does count the number of reduced residues $a$, modulo the product of the first $n$ primes except for those primes less than or equal to $k$, for which $a+1, \dots, a+k-1$ are all reduced residues. It's a rather specialized situation.

2: For a similar formula that counts $k$-tuples of reduced residues more generally, look at the Prime $k$-tuples Conjecture, particularly the infinite product of primes that appears as a constant in it.