Erdős Problem 854: erdos_854.parts.i

Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes.

If $1=a_1<a_2<\cdots <a_{\phi(n_k)}=n_k-1$ is the sequence of integers coprime to $n_k$, then estimate the smallest even integer not of the form $a_{i+1}-a_i$. Are there $\gg \max_i (a_{i+1}-a_i)$ many even integers of the form $a_{j+1}-a_j$?

This is the first part. A problem of Erdős [Er85c, p. 80], who writes "integer" instead of "even integer"; all differences $a_{i+1}-a_i$ are even, so only even integers are meaningful. Here "estimate" is read as $\Theta$. Erdős first thought that for large $k$ every even $t \leq \max_i(a_{i+1}-a_i)$ is a gap, but [Ob1] reports computations of Lacampagne and Selfridge showing this fails for $n_6 = 30030$ (OEIS A389839 and A048670 give $20$ and $22$).