Search for binomial coefficients $\binom{n}{k}$ equal to a product of consecutive primes (Erdős #386)

Call an integer a 'primorial-block' if it equals $p_i p_{i+1}\cdots p_j$, a product of consecutive primes each to the first power (e.g. $210=2\cdot3\cdot5\cdot7$). Erdős asked: for $2\le k\le n-2$, can $\binom{n}{k}$ be such a product of consecutive primes infinitely often? A known example is $\binom{21}{2}=210=2\cdot3\cdot5\cdot7$. GOAL (finite search): enumerate all pairs $(n,k)$ with $2\le k\le n/2$ and $n\le N$ for which $\binom{n}{k}$ is a product of consecutive primes, tabulate the solutions found, and report whether they thin out or persist. For each $(n,k)$ this is a decidable factorization/squarefree + consecutive-prime-block test.

Acceptance. PARTIAL / PRIMARY (finite, machine-checkable): a complete enumeration, for $n\le N$ (state $N$) and $2\le k\le n/2$, of all $(n,k)$ with $\binom{n}{k}$ a product of consecutive primes — delivered as the list of solutions plus a runnable script that factors each candidate (or applies Kummer/Legendre valuations) and verifies the consecutive-prime-block property, with exact arithmetic. Finding solutions with $k\ge 3$, or a run showing an unexpected density, is of direct interest. FULLY RESOLVES (not finite): a proof that infinitely many (or only finitely many) such $(n,k)$ exist. State $N$ and the factorization method.