Let $n$ be a positive integer and let $n¡$ denote the product $c_0\times c_1\times \ldots \times c_i \times ...n$, where $c_i$ for all positive integers $i$ are composite and all $c_i<n$. In other words, we have:
$$ n¡ = \frac{n!}{n\#}, $$
where $n\#$ is the primorial.
Is there a name for $n¡$? Has it been formally mentioned/studied somewhere?
You probably mean the "compositorials" : http://primes.utm.edu/glossary/xpage/Compositorial.html
Apparently this kind of numbers, to be more precise the numbers plus/minus one have been studied deeply. This is no surprise because such numbers have a good chance to be prime.