Is there a known means of measuring the biggest difference of consecutive elments of the reduced residue system modulo N?
For example, say we have the reduced residue system modulo 15: [1, 2, 4, 7, 8, 11, 13, 14]. The biggest difference between elements of this system is $7 - 4 = 11 - 8 = 3$. Is there a known way of calculating this directly knowing only the number our reduced residue system is calculated modulo?
I have tried and failed to identify such a function myself. I am interested in this because I want to study the relationships between these values for different reduced residue systems.
EDIT: Just to clarify, a reduced residue system modulo N can be constructed from a complete residue system modulo N by removing those elements of the complete residue system modulo N that are not relatively prime with N; the reduced residue systems this question considers are strictly those which are constructed from the complete residue system modulo N that is made up of by the interval of integers [1, N]. In other words, I am only interested in reduced residue systems in which every element of the system is smaller than N.
EDIT: It's been a few days, and I still haven't gotten an answer to this question. I take it that either I didn't ask the question very well, or no one knows of such a function. If it is the former, please comment and ask me to clarify.
This turns out to be a well-known problem that is difficult to solve: the answer to your question is equal to the Jacobsthal function $j(n)$.