Almost every text of number theory contains in its first chapters something similar to the following:
For any integer n, the factorial n! is the product of all positive integers up to and including n. Then in the sequence $\,n!+2,\,n!+3,\,\ldots,\,n!+n\,$ the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of $\,n-1\,$ consecutive composite integers, and it must belong to a gap between primes having length at least $\,n$.
But what about gaps between primes where each composite is divisible by increasing integers $\,(2,\,3,\,4,\,\ldots)$, without using factorials?
Obviously, we shall say that $\,g_n\,$ is a "maximal gap", if $\,g_m\lt g_n\,$ for all $\,m\lt n$.
A list of the primes bounding the first maximal gaps (found searching into the first $\,10^7$ primes):
$(g=2)\;\;\;\;\;\;\,3,\,5\;\;\;\;\;\;\;\;(2|4)$
$(g=4)\;\;\;\;\;\;13,\,17\;\;\;\;\;\,(2|14,\,3|15,\,4|16)$
$(g=6)\;\;\;\;\;\;\,61,\,67\;\;\;\;\;(2|62,\,3|63,\,4|64,\,5|65,\,6|66)$
$(g=10)\;\;\;\;\;2521,\,2531\;\;\;\;\;(\ldots)$
$(g=12)\;\;\;\;\;471241,\,471253\;\;\;\;\;(\ldots)$
$(g=16)\;\;\;\;\;4324321,\,4324337\;\;\;\;\;(\ldots)$
$(g=18)\;\;\;\;\;110270161,\,110270179\;\;\;\;\;(\ldots)$
Here, instead, is a list of the frequencies ($f_g$) of all gaps (also not maximal) up to the first $\,10^7$ primes:
$(g=2)\;\;\;\;\;\;\,f_2=738597$
$(g=4)\;\;\;\;\;\;\,f_4=368781$
$(g=6)\;\;\;\;\;\;\,f_6=123052$
$(g=10)\;\;\;\;\;f_{10}=4136$
$(g=12)\;\;\;\;\;f_{12}=447$
$(g=16)\;\;\;\;\;f_{16}=17$
$(g=18)\;\;\;\;\;f_{18}=1$
Some remarks: starting from $g=6$, in order to have satisfied the divisibility by 5, the first prime must be congruent to 1 (mod 20); $\,g\,$ seems always long $\,p-1$ (or it is just a coincidence?); $\,f_2\sim 2f_4\sim 6f_6$; maybe, there are infinitely many maximal gaps.
Could the previous remarks (except for the first) be explained in any way?
Many thanks.