I've been very interested at the gaps the between the elements of a reduced residue system modulo a primorial $p\#$.
The reason for this interest is that unlike the primes, the elements of reduced residue systems are easy to count using Euler's totient function and the pattern repeats so that I only need to consider a limited bound of $0 < x < p\#$.
When I do a search, I end up coming across questions that I have asked in the past. This is a wake up call that I am asking my question in a nonstandard way. I would greatly appreciate help in stating this question in a more standard way (or better understanding why this question is not interesting).
I've noticed that for the reduce residue system modulo $30$, each element fits neatly into an interval based on $\dfrac{30}{8} = 3.75$.
- $0 < 1 < 3.75$
- $3.75 < 7 < 7.5$
- $7.5 < 11 < 11.25$
- and so on.
When I did the same for $7\# = 210$, I counted $12$ times in $0 < x < 210$ where there was no element between $a\dfrac{210}{48}$ and $(a+1)\dfrac{210}{48}$.
I am at a loss where this number $12$ comes from. Is it random? Can it be calculated?
I am very interested how this number will increase for each successive prime. For example, what is this number for $11\# = 2310$ (how many times in $0 < x < 2310$, is there no element in the reduced residue system between $a\dfrac{2310}{480}$ and $(a+1)\dfrac{2310}{480}$?
Is there a standard mathematical way to analyze this question? Is there a clearer way for me to ask this question in a more standard way?